Free math worksheets

A plant is modeled as \(12\,\text{cm}\) tall at the start of an observation period and growing at a constant rate of \(1.5\,\text{cm}\) per week. Let \(h(w)\) be its height, in centimeters, after \(w\) weeks. Write a linear function for the model.

A swimming pool is being prepared for summer. At the start, there are already \(4\,\text{in.}\) of water in the pool. Water is then added at a constant rate. The table shows the water depth at different times: <table> <tr><td>Time \(t\) (h)</td><td>\(0\)</td><td>\(1\)</td><td>\(2\)</td><td>\(3\)</td><td>\(4\)</td><td>\(5\)</td></tr> <tr><td>Water depth \(h\) (in.)</td><td>\(4\)</td><td>\(10\)</td><td>\(16\)</td><td>\(22\)</td><td>\(28\)</td><td>\(34\)</td></tr> </table> a) Describe the graph formed by the data points, with time on the x-axis and water depth on the y-axis. b) How many inches per hour does the water depth increase? c) Write a function for the water depth \(h\) after \(t\) hours. d) Use your function to find the water depth after \(2.5\) hours.

A gym offers two monthly plans. Plan A costs a flat \(\$45.00\). Plan B costs \(\$10.00\) plus \(\$3.50\) per visit. Let \(v\) be the number of visits in a month. Write a cost function for each plan. Then evaluate this claim: “I work out every other day in a \(30\)-day month, so Plan B is definitely less expensive.” Justify your conclusion and state the difference in cost.

A line \(g\) is parallel to the x-axis and passes through \(A(-4, -7)\). 1. Find the equation of line \(g\). 2. For each point, determine whether it lies above, below, or on line \(g\): \(B(-4, -5)\), \(C(10, -7)\), and \(D(0, -10)\). Justify each answer by comparing y-coordinates.

In a spring-stretching experiment, \(x\) is the attached mass in grams and \(y\) is the spring's total length in centimeters. <table> <tr><td>Mass \(x\) (\(\text{g}\))</td><td>\(0\)</td><td>\(50\)</td><td>\(100\)</td><td>\(150\)</td><td>\(200\)</td></tr> <tr><td>Length \(y\) (\(\text{cm}\))</td><td>\(12.0\)</td><td>\(14.5\)</td><td>\(17.0\)</td><td>\(19.5\)</td><td>\(22.0\)</td></tr> </table> a) Find the linear function \(f(x) = mx + b\) that exactly models the data. b) Interpret the slope \(m\) and y-intercept \(b\) in the context of the experiment.

A candle is \(8\,\text{in.}\) tall when it is lit. It burns at a constant rate, and its height decreases by \(1\,\text{in.}\) each hour. a) Write a linear function for the candle's height \(h\), in inches, after \(t\) hours. b) After how many hours will the candle be half its original height? c) When will the candle be completely burned down?

When a freezer is turned on, its inside temperature is \(72\,^\circ\text{F}\). The temperature decreases at a constant rate of \(8\,^\circ\text{F}\) per hour. 1) Write a linear function for the temperature \(y\), in degrees Fahrenheit, after \(x\) hours. 2) Find the temperature after \(5.5\) hours. 3) After how many hours will the freezer reach its target temperature of \(0\,^\circ\text{F}\)?

Paul is saving for a new bike. He already has \(\$45.00\) saved, and he adds \(\$7.50\) each week. 1) Write a linear function for the amount \(y\), in dollars, after \(x\) weeks. 2) Find the points on the graph after \(0\), \(2\), \(4\), \(6\), and \(8\) weeks. 3) How many weeks will it take Paul to have exactly \(\$120.00\)?

The graphs of three linear functions \(f\), \(g\), and \(h\) are shown. a) Write an equation in the form \(y = mx + b\) for each line. b) Which function represents a proportional relationship? Justify your answer using both the graph and the equation. c) Describe how the three lines are positioned relative to one another. Explain how you can tell directly from their equations.

The graph shows two linear functions, \(f\) and \(g\). a) For \(f\), find the y-intercept \(b_1\) and slope \(m_1\). Then write the function equation. b) For \(g\), find the y-intercept \(b_2\) and slope \(m_2\). Then write the function equation.

At the start of a fire-department training exercise, a water tank contains \(3000\) gallons. Water is pumped out at a constant rate of \(375\) gallons per hour. The graph shows the amount of water remaining as a function of time. a) Use the graph to estimate how many hours pass before \(1125\) gallons remain. b) Write a linear function \(V\), where \(t\) is time in hours and \(V(t)\) is the remaining volume in gallons. State the contextual domain, and use the function to find exactly when the tank is empty.

The figure shows two linear functions, \(f\) and \(g\). Two points are marked on each line. For each line, find the slope \(m\), the y-intercept \(b\), and the complete equation in the form \(y = mx + b\).

Find the equation of each line \(f\), \(g\), and \(h\) shown in the coordinate plane. Write each equation in the form \(y = mx + b\).

The line \(g\) shown in the coordinate plane passes through points \(P_1\) and \(P_2\). a) Use the coordinates of \(P_1\) and \(P_2\) to calculate the slope \(m\). b) Calculate the y-intercept \(b\). c) Write the complete function equation in the form \(g(x) = mx + b\).

The coordinate plane shows four lines, \(g_1\) through \(g_4\). Find the equation of each line. What do you notice when you compare the equations?

Three lines, \(f\), \(g\), and \(h\), are shown. Write an equation for each line. How can you tell from the equations that the lines are parallel?

The coordinate plane shows three lines, \(f\), \(g\), and \(h\). Find the equation of each line.

Consider the line \(g\) shown in the coordinate plane. a) Find the y-intercept \(b\). b) Find the slope \(m\). c) Write the equation in the form \(y = mx + b\).

Find the equation of each line \(f\) and \(h\) shown in the coordinate plane. Use the y-intercept and a slope triangle.

The graph shows four lines, \(h_1\) through \(h_4\). Find the equation of each line.

A storage tank contains \(10\,\text{gal}\) of water at the start. It is then filled at a constant rate. The graph shows the volume \(V\), in gallons, after \(t\) minutes. The axes use different scales. a) Find the linear function \(V(t)\). b) Find the amount of water in the tank after \(6\) minutes. Then check your result on the graph.

At the start of a \(15\)-minute observation, a rainwater tank already contains \(25\) gallons. A constant inflow adds \(5\) gallons per minute. The graph shows the tank's volume as a function of time. a) Use the graph to determine how many minutes pass before the tank contains \(60\) gallons. b) Write the function, including its domain for the observation interval, and find when the tank contains \(90\) gallons.

A new candle is \(12\) inches tall. As it burns, its height decreases at a constant rate of \(0.75\) inch per hour. The graph shows candle height over time. a) Use the graph to find the candle's height after \(4\) hours. b) Write the function, including its contextual domain, and find how many hours pass before the candle is \(4.5\) inches tall.

Two hikers, Alex and Bea, travel along a long straight road. Alex starts at mile marker \(0\) and walks at a constant speed toward higher mile markers. Bea starts at the same time at mile marker \(10\) and walks at a constant speed toward mile marker \(0\). On the graph, curve \(a\) represents Alex and curve \(b\) represents Bea. Write a function of the form \(s(t)=mt+b\) for each hiker on the displayed interval \(0\le t\le4\).

Find the equations of lines \(a\) and \(b\). Pay special attention to the direction of line \(a\).

Two water tanks are observed at the same time. The graph shows volume \(V\), in gallons, as a function of time \(t\), in minutes. Curve \(a\) represents Tank 1, and curve \(b\) represents Tank 2. Write an equation of the form \(V(t)=mt+b\) for each tank, including the displayed domain.

A taxi company charges a fixed starting fee plus a constant price per mile. The graph shows the total cost \(y\), in dollars, for a trip of \(x\) miles. Find the equation of the line.

Two runners, Alex and Blake, train on a long, straight trail. Alex starts at mile marker \(0\) and runs at a constant speed of \(8\,\text{mi/h}\). Blake starts at the same time at mile marker \(2\) and runs at a constant speed of \(6\,\text{mi/h}\). Write a linear position function for each runner, then use the models to determine when and where Alex catches Blake.

A smartphone battery drains at a constant rate during a certain type of use. It is fully charged at \(10{:}00\) a.m. The table shows the battery level at several times. <table><tr><td>Time</td><td>\(10{:}00\) a.m.</td><td>\(12{:}00\) p.m.</td><td>\(2{:}00\) p.m.</td><td>\(3{:}00\) p.m.</td><td>\(6{:}00\) p.m.</td></tr><tr><td>Battery level (%)</td><td>\(100\)</td><td>\(84\)</td><td>\(68\)</td><td>\(60\)</td><td>\(36\)</td></tr></table> a) Find the average decrease in battery level per hour. Verify that the rate is constant over every interval in the table. b) Estimate the battery level at \(4{:}30\) p.m. Explain your method. c) If the pattern continues, at what time will the battery reach \(0\%\)? d) Write a rule for the battery level \(y\) after \(x\) hours have passed since \(10{:}00\) a.m.

A taxi company charges a base fare of \(\$4.50\) plus \(\$2.20\) per mile. a) Write an equation for the total fare \(y\), in dollars, after traveling \(x\) miles. b) Explain what \(4.50\) represents in this situation. c) A passenger pays exactly \(\$26.50\). How many miles did the passenger travel?

Two mobile data plans have the following monthly costs. - Plan A: a \(\$10.00\) base fee plus \(\$0.50\) per gigabyte. - Plan B: no base fee and \(\$1.50\) per gigabyte. a) Write an expression for each plan's monthly cost \(C\) as a function of data use \(x\), in gigabytes. b) Find both costs for \(5\,\text{GB}\) and \(15\,\text{GB}\). c) For what data use is Plan A less expensive than Plan B? Explain by comparing the expressions or testing values.

A candle is \(30\,\text{cm}\) tall when it is lit. After burning at a constant rate for \(4\,\text{hours}\), it is \(22\,\text{cm}\) tall. a) Is the relationship from burn time (in hours) to remaining height (in centimeters) proportional? Briefly explain. b) Write a function \(h(x)\) for the candle's remaining height after \(x\) hours, while the candle is burning. c) Use your function to find the candle's height after \(7.5\,\text{hours}\).

Two different candles are lit at the same time. Candle A is initially \(30\,\text{cm}\) tall and is \(18\,\text{cm}\) tall after \(4\) hours. Candle B is initially \(20\,\text{cm}\) tall and burns down at a constant rate of \(1\,\frac{\text{cm}}{\text{h}}\). Let \(t\) be the number of hours after the candles are lit. a) Determine Candle A's constant rate of change and construct linear height functions \(h_A(t)\) and \(h_B(t)\). b) Use your model to find Candle A's height after \(7\) hours and the time when Candle A reaches height \(0\). c) Use the two functions to determine when the candles have the same height and what that height is.

Lara watches her smartphone battery during a long train ride while playing a game continuously. At \(12{:}15\) p.m., the battery level is \(88\%\). At \(2{:}45\) p.m., it is \(63\%\). a) Find the decrease in battery level, in percentage points per hour. Under this linear assumption, at what time would the battery reach \(0\%\)? b) According to the model, what would the battery level have been at \(10{:}15\) a.m.? c) Explain why the result from part b shows that the linear model cannot be valid for the entire morning.

A sports club tracks membership in its tennis and basketball programs. <table> <thead> <tr><th>Year</th><th>2010</th><th>2013</th><th>2016</th><th>2019</th><th>2022</th></tr> </thead> <tbody> <tr><td>Tennis</td><td>\(240\)</td><td>\(230\)</td><td>\(215\)</td><td>\(205\)</td><td>\(190\)</td></tr> <tr><td>Basketball</td><td>\(80\)</td><td>\(110\)</td><td>\(145\)</td><td>\(175\)</td><td>\(210\)</td></tr> </tbody> </table> For parts a)–c), assume each program changed at a constant rate from 2019 onward equal to its average rate from 2019 to 2022. Let \(t\) be the number of years after 2019. a) Construct linear models \(T(t)\) and \(B(t)\) for tennis and basketball membership. b) Use the models to estimate when the memberships were equal. c) Use the basketball model to predict membership in 2025.

A pump fills an empty tank at a constant rate. After \(15\) minutes, the tank contains \(450\) gallons. a) Write a function for the volume \(V\), in gallons, after \(t\) minutes. b) A second pump moves twice as much water per minute. Write its function. c) How long would each pump take to fill a \(2700\)-gallon tank? How much time does the faster pump save?

A vehicle uses \(18\) gallons of gasoline to travel \(600\) miles. Assume gasoline used is proportional to distance. a) Write a function of the form \(f(x)=kx\) for the gasoline use \(f(x)\), in gallons, after traveling \(x\) miles. What does \(k\) mean in context? b) While towing a trailer, the vehicle uses an additional \(1.2\) gallons for every \(100\) miles. How many gallons will it use under these conditions on a \(350\)-mile trip?

A class is ordering printed T-shirts. InkWorks charges a one-time setup fee of \(\$24\) plus \(\$7\) per shirt. TeeDirect charges \(\$10\) per shirt with no additional fee. Let \(x\) be the number of shirts ordered. a) Construct cost functions \(I(x)\) and \(T(x)\) for InkWorks and TeeDirect. b) Use the functions to find the cost of \(5\) shirts and \(10\) shirts from each company. c) Starting with how many whole shirts is InkWorks less expensive than TeeDirect?

Lucas and Sarah are saving for bicycles. Lucas already has \(\$60\) and adds \(\$4\) each week. Sarah has \(\$10\) and adds \(\$9\) each week. a) Write a linear model for each person's savings after \(w\) weeks. b) Use the models to determine after how many weeks they will have the same amount and state that amount.

A large tank contains \(450\,\text{L}\) of water and loses \(15\,\frac{\text{L}}{\text{min}}\) through a leak. At the same time, a second tank, initially empty, is filled at \(25\,\frac{\text{L}}{\text{min}}\). Let \(t\) be the number of minutes after both processes begin. a) Construct volume functions \(V_1(t)\) and \(V_2(t)\) for the two tanks. b) Use the functions to find when the tanks contain the same amount of water and how much each contains then. c) Explain from the two models why the second tank contains more water after that time, while both models remain physically meaningful.

Two conveyor belts add boxes to two stacks. Stack A starts with \(15\) boxes and receives \(5\) boxes per minute. Stack B starts with \(45\) boxes and receives \(2\) boxes per minute. Let \(t\) be the number of minutes after the belts start. a) Construct functions \(A(t)\) and \(B(t)\) for the number of boxes in each stack. b) Which stack grows faster? Explain using the functions' rates of change. c) Use the functions to find when the stacks contain the same number of boxes and how many boxes each contains then.

Two candles are lit at the same time. Candle A is \(20\,\text{cm}\) tall and burns down \(1.5\,\frac{\text{cm}}{\text{h}}\). Candle B is \(15\,\text{cm}\) tall and burns down \(0.5\,\frac{\text{cm}}{\text{h}}\). Let \(t\) be the number of hours after the candles are lit. a) Construct height functions \(A(t)\) and \(B(t)\). b) Use the functions to find when the candles have the same height and what that height is. c) Which candle is taller after \(7\) hours? Justify using the functions.

Julian compares two monthly cell phone plans. The Basic plan costs \(\$12.00\) plus \(\$0.08\) per minute. The Flat plan costs \(\$22.00\), regardless of the number of minutes used. Let \(m\) be the number of minutes used in a month. a) Construct monthly cost functions \(B(m)\) and \(F(m)\) for the Basic and Flat plans. b) Use the functions to determine the least whole number of minutes for which the Flat plan is less expensive than the Basic plan. Show the comparison as an inequality.

Luke compares two gym memberships. Power: a \(\$45.00\) sign-up fee plus \(\$25.00\) per month Easy: no sign-up fee and \(\$35.00\) per month Let \(m\) be the number of months of membership. a) Construct total-cost functions \(P(m)\) and \(E(m)\). b) Use the functions to find the total cost of each membership after \(4\) months. c) Starting with which full month is Power less expensive overall than Easy? Justify with a comparison of the two functions.

A family compares two annual cell phone plans. Smart: a \(\$114.00\) annual fee plus \(\$0.05\) per minute Basic: no annual fee and \(\$0.15\) per minute Let \(m\) be the number of minutes used during the year. a) Construct annual cost functions \(S(m)\) and \(B(m)\). b) Compare the annual costs for \(960\) minutes. c) Starting at how many whole minutes per year is Smart less expensive than Basic?

Two car-sharing companies offer these monthly plans. Company A: a \(\$15.00\) monthly fee plus \(\$0.20\) per mile Company B: no monthly fee and \(\$0.45\) per mile Let \(m\) be the number of miles driven in a month. a) Construct monthly cost functions \(A(m)\) and \(B(m)\). b) Use the functions to find the monthly cost for each company at \(40\) miles. c) For what monthly mileage is Company A less expensive than Company B?

A portable solar panel charges a battery. Under constant conditions, the stored energy increases by \(4\) watt-hours, or \(\text{Wh}\), every \(15\) minutes. Assume the battery starts empty and can store at least \(50\,\text{Wh}\). a) How much energy is stored after \(2\) hours? b) After how many hours, minutes, and seconds has the panel delivered exactly \(50\,\text{Wh}\)? c) Write a function in the form \(y=kx\), where \(x\) is time in minutes and \(y\) is energy in watt-hours. Explain what \(k\) means in context.

A vehicle uses gasoline at a constant rate of \(4\) gallons per \(100\) miles. a) Write a function for the volume \(V\), in gallons, used after traveling \(s\) miles. b) Find the gasoline used on a \(450\)-mile trip. c) The tank has \(12\) gallons remaining. How far can the vehicle theoretically travel?

On a sunny day, Anja observes the shadow of a vertical pole in the schoolyard. She marks the shadow's position at \(10{:}00\) a.m. At \(12{:}30\) p.m., the shadow has swept through an angle of \(37.5^\circ\). Assume the shadow angle changes at a constant rate from \(10{:}00\) a.m. through \(4{:}15\) p.m. a) Find the change in angle per hour. b) Write a function for the angle \(\alpha\), in degrees, after \(t\) hours from the first measurement. c) What total angle has the shadow swept through by \(4{:}15\) p.m.?

A linear function \(g\) has graph passing through \(P(-1,7)\) and \(Q(2,-2)\). a) Construct the function in the form \(g(x)=mx+b\). b) Use your function to determine whether \(R(0,5)\) and \(S(1,1)\) are input-output pairs of \(g\).

An electric-vehicle charging plan has a monthly base fee of \(\$18.50\) plus \(\$0.80\) for each kilowatt-hour (\(\text{kWh}\)) used. Let \(x\) be the number of kilowatt-hours used in a month. a) Construct a monthly cost function \(C(x)\), and explain what its initial value and rate of change represent. b) A customer says a month with \(40\,\text{kWh}\) of charging cost exactly \(\$50.50\). Does the function support the claim? c) How many kilowatt-hours were used if the monthly bill was exactly \(\$74.50\)?

A line \(g\) passes through \(P(4, -1)\) and has slope \(m = -0.75\). A student wants to find more points on the line using slope triangles. a) Starting at \(P\), the student moves 8 units left. How many units must the student move up or down to land on the line again? b) What is the vertical change when the student moves exactly 1 unit right? c) Find an equation for line \(g\).

A small backyard pool already contains \(30\) gallons of water. A garden hose adds water at a constant rate of \(4\) gallons per minute. a) Write a function \(V(t)\) for the volume of water, in gallons, after \(t\) minutes. b) How much water is in the pool after \(20\) minutes? c) How long will it take for the pool to contain \(150\) gallons?

Two candles are lit at the same time. Candle A starts at \(24\,\text{cm}\) and burns down at \(2\,\text{cm/h}\). Candle B starts at \(18\,\text{cm}\) and burns down at \(1.2\,\text{cm/h}\). a) Write a function for the height \(h\), in centimeters, of each candle after \(t\) hours. b) Which candle is taller after \(5\) hours? Show your calculation. c) After how many hours will the two candles have the same height?

A delivery service charges a fixed fee plus a constant amount per mile. A \(10\)-mile delivery costs \(\$17.00\), and a \(25\)-mile delivery costs \(\$32.00\). a) Write a function \(C(d)\) for the total cost, in dollars, of a delivery that travels \(d\) miles. b) What is the fixed fee? c) Find the cost of a \(40\)-mile delivery.

A candle burns at a constant rate. After \(3\) hours, it is \(7\) inches tall. Its height decreases by \(0.75\) inch each hour. a) Write a function \(h(t)\) for the candle's height, in inches, after \(t\) hours. b) How tall was the candle when it was first lit?

A candle burns at a constant rate. Two hours after it is lit, it is \(7\) inches tall. After \(5\) hours, it is \(4\) inches tall. 1. Write a function \(f(x)\) for the candle's height, in inches, after \(x\) hours. 2. How tall was the candle when it was lit? 3. After how many hours will the candle be completely burned down? 4. What domain for \(x\) makes sense in this context?

Water flows into a large tank at a constant rate. After \(10\) minutes, the tank contains \(110\) gallons. After \(25\) minutes, it contains \(200\) gallons. 1. Write a function \(V(x)\) for the volume of water, in gallons, after \(x\) minutes. 2. How much water was in the tank when the timing began? 3. The tank holds \(290\) gallons. After how many minutes will it be full?

A rain barrel contains \(80\) gallons of water at time \(t = 0\). Water drains from an open spigot at a constant rate. After \(40\) minutes, \(60\) gallons remain. a) Write a linear function \(V(t)\) for the amount of water, in gallons, after \(t\) minutes. b) After how many minutes will the barrel be empty?

Two water tanks are observed at the same time. Tank A starts with \(500\,\text{L}\) and is drained at a constant rate of \(12\,\text{L/min}\). Tank B starts with \(200\,\text{L}\) and is filled at a constant rate of \(8\,\text{L/min}\). Write a function for the volume of each tank after \(t\) minutes. Then find when the tanks contain the same amount of water and what that common volume is.

A candle is \(10\) inches tall when it is lit. After \(3\) hours, it is \(8.5\) inches tall. a) Write a function \(h(t)\) for the candle's height, in inches, after \(t\) hours. b) Explain what the slope \(m\) and y-intercept \(b\) mean in this situation. c) Find the candle's height after \(5\) hours.

A water tank fills at a constant rate. After \(10\) minutes, it contains \(350\) gallons. After \(25\) minutes, it contains \(800\) gallons. a) Write a linear function \(V(t)\) for the volume, in gallons, after \(t\) minutes. b) How much water was in the tank at \(t = 0\)? c) The tank holds \(1700\) gallons. After how many minutes will it be full?

A taxi company uses a linear fare. A \(5\)-mile ride costs \(\$14.50\), and a \(12\)-mile ride costs \(\$28.50\). a) Write a function \(C(x)\) for the fare, in dollars, for a ride of \(x\) miles. b) Explain what the slope \(m\) and y-intercept \(b\) mean in this situation. c) Find the fare for a \(20\)-mile ride.

Air temperature in the atmosphere can be modeled as decreasing linearly with altitude. At \(2000\) feet above sea level, a meteorologist measures \(50^\circ\text{F}\). At \(10{,}000\) feet, the temperature is \(18^\circ\text{F}\). a) Write a linear function \(T(h)\) for the temperature, in degrees Fahrenheit, at altitude \(h\) feet. b) What temperature does the model predict at sea level? c) At what altitude does the model predict the freezing temperature of \(32^\circ\text{F}\)?

Two monthly phone plans are available. The Basic plan charges a monthly fee of \(\$8.00\) plus \(\$0.15\) per calling minute. The Smart plan charges a monthly fee of \(\$14.00\) plus \(\$0.05\) per calling minute. a) Write a function for the monthly cost \(y\) of each plan in terms of the number of calling minutes \(x\). b) Find the number of calling minutes for which the two plans have the same monthly cost. c) Explain which plan is less expensive for lighter use and which is less expensive for heavier use.

Two hiking groups are on the same trail to a mountain summit. Group A starts at a parking lot and hikes at a constant speed of \(3\,\text{mi/h}\). Group B starts at the same time from a lodge \(2\) miles farther along the trail and hikes at \(2\,\text{mi/h}\). a) Write a function for each group's distance \(s\), in miles, from the parking lot after \(t\) hours. b) Find when and where Group A catches Group B. c) The summit is \(7\) miles from the parking lot. Determine mathematically whether Group A catches Group B before either group reaches the summit.

A tablet battery is charging. At the start, the battery level is \(10\%\). After \(24\) minutes, it has increased at a constant rate to \(40\%\). a) Write a linear function \(y\) for the battery level, in percent, after \(x\) minutes. b) Assuming the charging rate stays constant, after how many minutes will the battery reach \(100\%\)? c) Describe the meaningful graph of the model by stating its two endpoints and whether it is increasing, decreasing, or constant. You may sketch it on paper to help.

A rectangular backyard pool is filled at a constant rate. At \(1{:}00\) p.m., the water is \(16\,\text{in.}\) deep. At \(3{:}00\) p.m., it is \(40\,\text{in.}\) deep. Let \(t\) be the number of hours after \(1{:}00\) p.m., and let \(d(t)\) be the water depth in inches. a) Construct a linear depth function \(d(t)\) from the two measurements. State its rate of change. b) Extending the same model backward, at what clock time does \(d(t)=0\)? c) The pool is full at a depth of \(64\,\text{in.}\). At what clock time does the model reach that depth?

A vehicle travels at a constant speed during a highway test. The table shows time \(t\), in hours, and distance \(s\), in miles. <table> <tr><td>Time \(t\), in hours</td><td>\(0.5\)</td><td>\(1.5\)</td><td>\(2\)</td></tr> <tr><td>Distance \(s\), in miles</td><td>\(30\)</td><td>\(90\)</td><td>\(120\)</td></tr> </table> a) Use calculations to determine whether the table could represent a proportional relationship. b) State the physical condition that must hold throughout the trip for this proportionality to be exact. c) Write the function for the relationship.

Two electricity providers offer different monthly plans. Provider A: no monthly base charge and \(\$0.40\) per kilowatt-hour, or \(\text{kWh}\). Provider B: a \(\$10.00\) monthly base charge and \(\$0.30\) per kilowatt-hour. a) Write a function for monthly cost \(y\), in dollars, as a function of usage \(x\), in kilowatt-hours, for each provider. b) Which plan represents a proportional relationship? Justify your answer mathematically. c) How does Provider B's base charge change its graph compared with a line through the origin?

Two water tanks are being adjusted at the same time. Tank 1 starts with \(2500\,\text{L}\) of water and loses \(150\,\text{L}\) per hour. Tank 2 starts with \(500\,\text{L}\) of water and gains \(250\,\text{L}\) per hour. a) Write a function for the volume \(V\), in liters, of each tank after \(t\) hours. b) Find when the two tanks contain the same amount of water and determine that common volume. c) Tank 2 has a maximum capacity of \(3000\,\text{L}\). How much water remains in Tank 1 when Tank 2 becomes full?

A candle is \(9.5\) inches tall when it is lit. After \(4\) hours, it is \(5.5\) inches tall. Its height decreases at a constant rate. a) Write a function \(h(t)\) for the candle's height, in inches, after \(t\) hours. b) After how many hours and minutes will the candle be completely burned down?

A liquid cools at a constant rate. After \(2\) minutes, its temperature is \(170^\circ\text{F}\). After \(10\) minutes, its temperature is \(130^\circ\text{F}\). a) Write a linear function \(T(t)\) for the temperature, in degrees Fahrenheit, after \(t\) minutes. b) Explain what the slope and y-intercept mean in this situation.

A water container fills at a constant rate. After \(5\) minutes, it contains \(5.5\) gallons. After \(12\) minutes, it contains \(9\) gallons. a) Write a function \(V(t) = mt + b\) for the volume, in gallons, after \(t\) minutes. b) How much water was in the container at time \(0\)? c) How much water will be in the container after half an hour?

A prepaid phone account starts with a balance of \(\$20.00\). Each minute of calling costs \(\$0.12\). a) Write a function \(G(t)\) for the remaining balance, in dollars, after \(t\) minutes of calls. b) How much money remains after \(45\) minutes of calls? c) After how many minutes will exactly \(\$5.00\) remain? d) Is this relationship proportional? Explain.

Two trains travel in the same direction on parallel tracks. A local train travels at a constant speed of \(60\,\text{mi/h}\). An express train travels at \(80\,\text{mi/h}\). At the start of the observation, the local train is \(2\) miles ahead. a) Write a function for each train's distance \(s\), in miles, after \(t\) minutes. b) After how many minutes does the express train catch the local train? c) How far has the express train traveled from the start of the observation by that time?

A school is comparing two online photo-printing offers. PhotoNow charges \(\$0.12\) per photo plus a flat \(\$3.50\) shipping fee. PrintPlus charges \(\$0.17\) per photo plus a flat \(\$1.50\) shipping fee. Write a function for the total cost \(y\) of each offer for \(x\) photos. Then find the number of photos for which the two offers cost exactly the same amount.

A candle burns at a constant rate. Two hours after it is lit, it is \(7.5\) inches tall. After \(5\) hours, it is \(6\) inches tall. a) Write a function \(h(t)\) for the candle's height, in inches, after \(t\) hours. b) Explain what the y-intercept and slope mean in this situation.

A point \(P(x, y)\) lies on a line described by this rule: “The y-value is \(4\) more than twice the x-value.” a) Write a linear equation that represents the rule. b) Determine algebraically whether \(A(3, 10)\) and \(B(-2, 1)\) lie on the line. c) A point \(C\) on the line has y-coordinate \(15\). Find its x-coordinate.

Two climbing gyms offer different monthly pricing plans. Summit Climbing Gym charges a \(\$10\) monthly fee plus \(\$5\) per visit. Peak Climbing Gym has no monthly fee and charges \(\$7.50\) per visit. Let \(v\) be the number of visits in a month. a) Construct monthly cost functions \(S(v)\) and \(P(v)\). b) Use the functions to find the number of visits for which the plans cost exactly the same amount. c) For \(5\) visits, which plan costs less and by how much? d) Starting at how many whole visits per month is Summit less expensive overall?

Two water tanks are being adjusted at the same time. Tank A starts with \(100\) gallons of water, and \(15\) gallons are added each minute. Tank B starts with \(400\) gallons of water, and \(10\) gallons are drained each minute. a) Write a function for the volume \(y\), in gallons, in each tank after \(x\) minutes. b) Explain what the slope and y-intercept of the function for Tank B mean in this situation. c) Find when the tanks contain the same amount of water and how much water each tank contains at that time.

Two siblings, Liam and Sarah, are saving money. Liam starts with \(\$20\) and saves another \(\$5\) each week. Sarah starts with \(\$50\) but saves only \(\$2\) each week. a) Write a function for each person's savings \(y\) after \(x\) weeks. b) After how many weeks will they have the same amount of money? c) How much money will each person have at that time? d) Interpret the slope of each function in this situation.

An electric scooter rental offers two pricing plans: Plan A: a \(\$1.20\) activation fee plus \(\$0.20\) per minute Plan B: no activation fee and \(\$0.30\) per minute a) Write a cost rule for each plan for a ride lasting \(x\) minutes. b) Find the cost of a \(15\)-minute ride under each plan. Which plan is less expensive? c) Find the ride length for which the two plans cost the same.

A technician records the temperature of a cooling liquid: <table> <tr> <td>Time \(t\) in \(\text{min}\)</td> <td>\(0\)</td><td>\(8\)</td><td>\(20\)</td> </tr> <tr> <td>Temperature \(T\) in \(^\circ\text{C}\)</td> <td>\(90\)</td><td>\(74\)</td><td>\(50\)</td> </tr> </table> a) Show using calculations that the temperature can be modeled by a linear function over this time interval. b) Find a linear model of the form \(T(t) = mt + b\). c) Find the temperature after exactly \(15\) minutes. d) Assuming the linear pattern continues, when will the liquid reach \(26\,^\circ\text{C}\)?

A cylindrical container has an inside diameter of \(12\,\text{cm}\) and a height of \(30\,\text{cm}\). Water is poured into the container. a) Write a function \(V\) that gives the water volume \(V(h)\), in \(\text{cm}^3\), for a water height \(h\), in centimeters. b) State the domain \(D\) and range \(R\) for this situation. Round the upper endpoint of the range to the nearest hundredth.

A right triangular prism has a base with legs \(6\,\text{cm}\) and \(8\,\text{cm}\). The prism is \(25\,\text{cm}\) tall and stands on its triangular base while it is filled with liquid. a) Write a function \(f\) that gives the liquid volume \(f(x)\), in \(\text{cm}^3\), for a fill height \(x\), in centimeters. b) Find the fill height when the prism contains exactly \(0.36\,\text{L}\) of liquid. c) Briefly explain why \(f\) is a linear function.

A rain barrel initially contains \(10\) gallons of water and has a capacity of \(50\) gallons. During a storm, water flows into the barrel at a constant rate of \(1\) gallon per minute. 1) Write a function for the amount of water \(y\), in gallons, after \(x\) minutes, including the physical domain. 2) Make a table for \(x=0\), \(x=10\), \(x=15\), and \(x=30\). 3) Describe the graph of the function on its physical domain. 4) Determine algebraically when the barrel becomes full.

Two water tanks begin draining at the same time. Tank 1 starts with \(500\,\text{gal}\) and loses \(10\,\frac{\text{gal}}{\text{min}}\). Tank 2 starts with \(800\,\text{gal}\) and loses \(30\,\frac{\text{gal}}{\text{min}}\). Let \(t\) be the number of minutes after draining begins. a) Construct volume functions \(V_1(t)\) and \(V_2(t)\) for the two tanks while each model is physically meaningful. b) Use the functions to determine when Tank 1 contains exactly twice as much water as Tank 2. Verify that both modeled volumes are nonnegative at that time.

Two monthly cell phone plans are available. Plan A: a \(\$10\) monthly fee plus \(\$0.15\) per minute Plan B: no monthly fee and \(\$0.25\) per minute Let \(m\) be the number of minutes used in a month. a) Construct monthly cost functions \(A(m)\) and \(B(m)\). b) Use the functions to find the number of minutes for which the plans have the same monthly cost. c) Which plan costs less for \(150\) minutes? Justify using your functions.

Lena leaves for a hike at \(9{:}00\) a.m. and walks at a constant speed of \(3\) miles per hour. At \(10{:}30\) a.m., her brother Max realizes that she forgot her water bottle. He immediately rides his bike after her along the same route at \(12\) miles per hour. a) Write a function for each person's distance \(y\), in miles, from the starting point after \(x\) hours since Lena left. b) Identify two points on each person's graph and explain what an intersection of the graphs represents. c) Determine what time Max catches Lena and how far they are from the starting point. d) Explain what the slope of each line means in this situation.

Towns \(A\) and \(B\) are \(300\) miles apart along a straight rail line. A freight train leaves Town \(A\) traveling toward Town \(B\) at a constant speed of \(60\) miles per hour. At the same time, a passenger train leaves Town \(B\) traveling toward Town \(A\) at \(90\) miles per hour. a) Write a function for each train's distance \(y\), in miles, from Town \(A\) after \(x\) hours. Identify two points on each function. b) Compare the trains' distances at whole-number times to estimate when and where they meet. c) Calculate the exact meeting time and distance to check your estimate.

A smartphone battery is at \(15\%\) when charging begins. After \(20\) minutes, it is at \(45\%\). Assume the battery percentage increases linearly with time. a) Write a linear function for the battery percentage \(y\) after \(x\) minutes. b) How many percentage points are added per minute? Interpret this value as the slope. c) According to the model, how many minutes after charging begins will the battery reach \(100\%\)?

A smartphone battery is at \(12\%\) when it is connected to a charger that increases the battery level at a constant rate of \(0.8\) percentage point per minute. a) Write a linear function for the battery level \(L\), in percent, after \(t\) minutes. b) A fast charger starts at the same battery level and reaches \(100\%\) in exactly \(80\) minutes. Find its charging rate in percentage points per minute. c) How many minutes does the fast charger save compared with the first charger when charging from \(12\%\) to \(100\%\)?

A hot-air balloon is \(2800\,\text{ft}\) above the ground when it begins to descend at a constant rate of \(5\,\text{ft/s}\). 1) Write a linear function for the balloon's height \(h\), in feet, after \(t\) seconds. 2) How high is the balloon after \(4\) minutes of descending? 3) When will the balloon reach the ground? Give the time in minutes and seconds.

Two water tanks begin filling at the same time. Tank A starts with \(150\) gallons of water and receives another \(20\) gallons each minute. Tank B starts with only \(50\) gallons but fills at a rate of \(32.5\) gallons per minute. a) Write a function for the volume \(y\), in gallons, in each tank after \(x\) minutes. b) Find how many minutes it takes for the tanks to contain the same amount of water. c) How much water is in each tank at that time?

In a proportional relationship between \(x\) and \(y\), every increase of \(4\) in \(x\) produces an increase of \(14\) in \(y\). a) Write the equation in the form \(y=kx\). b) What x-value gives \(y=49\)?

A smartphone battery drains at a constant rate while the phone is in use. At \(10{:}00\) a.m., the battery level is \(80\%\). At \(11{:}20\) a.m., it is \(60\%\). a) Find the battery loss in percentage points per hour. b) Assuming the phone has been used continuously at this rate, at what time was it unplugged at \(100\%\)? c) If the same rate continues, at what time will the battery reach \(0\%\)?

Towns A and B are \(180\) miles apart along a highway. At \(12{:}00\) p.m., a truck leaves Town A and drives toward Town B at a constant speed of \(60\) miles per hour. At \(12{:}45\) p.m., a car leaves Town B and drives toward Town A at \(90\) miles per hour. Let \(t\) be the number of hours after \(12{:}00\) p.m., and measure position as distance from Town A. a) Construct the truck's position function \(T(t)\). For \(t\ge0.75\), construct the car's position function \(C(t)\). b) Use the two position functions to determine when the vehicles meet and the distance of the meeting point from Town A.

A large water tank is being drained at a constant rate. The graph shows the remaining volume \(V\), in gallons, after \(t\) minutes. a) Choose two convenient points from the graph and use them to find the slope \(m\). Explain what the slope means in this situation. b) Find the y-intercept \(b\) from the graph and write the linear function in the form \(V(t) = mt + b\). c) After how many minutes will the tank be empty?

The coordinate plane uses different scales on the x-axis and y-axis. a) Find the equation of the linear function \(g\) shown in the graph. b) Calculate \(g(6)\), then check your result by reading the corresponding point from the graph.

Maya and Noah ride bicycles along the same route. Maya leaves the starting point at \(10{:}00\) a.m. Noah leaves the same point later but rides faster. The graph labels Maya's distance function \(f\) and Noah's distance function \(g\). a) Who rides faster? Find each rider's speed in miles per hour. b) Use the graph to determine when Noah catches Maya and how far they have traveled by then. c) Write an equation for each rider's distance \(y\), in miles, as a function of clock time \(x\), measured in hours. Solve the system to find the intersection, and compare it with your answer to part b).

A water tank is filled at a constant rate. The function \(V(t)\) gives the volume in gallons after \(t\) minutes. Two measurements, \(P_1\) and \(P_2\), are shown on the graph. a) Find the inflow rate, or slope, in gallons per minute. b) Find the volume already in the tank at \(t=0\). c) Write the function \(V(t)\). d) Find the volume after \(15\) minutes.

Two cell phone plans have different monthly fees and different per-minute charges. Let \(x\) be the number of minutes used in a month. On the graph, curve \(a\) represents Plan A and curve \(b\) represents Plan B. Write a cost function for each plan, including the contextual domain. What does the intersection of the two lines mean in context?

Use the graph to write an equation for the line \(f\). Then read the zero, or x-intercept, from the graph and verify it algebraically.

Liam leaves a trailhead at \(9{:}00\) a.m. and walks at a constant speed of \(4\) miles per hour. At \(10{:}30\) a.m., Marie rides after him along the same route at \(12\) miles per hour. The graph shows each person's distance from the trailhead as a function of time \(t\), in hours since \(9{:}00\) a.m. The graph uses \(l\) for Liam and \(m\) for Marie. a) Construct the two distance functions \(l(t)\) and \(m(t)\) from the situation. State the time restriction on Marie's function. b) Use the graph to estimate when and how far from the trailhead Marie catches Liam. c) Use the functions to calculate the meeting time and distance exactly, and give the clock time.

Two cargo boats, the Albatross and the Osprey, travel along a \(100\)-mile route between two harbors. The Albatross leaves Harbor A toward Harbor B at \(20\) miles per hour. At the same time, the Osprey leaves Harbor B toward Harbor A at \(30\) miles per hour. The graph shows each boat's distance from Harbor A as a function of travel time \(t\). The graph uses \(a\) for the Albatross and \(o\) for the Osprey. a) Construct position functions \(a(t)\) and \(o(t)\), measuring distance from Harbor A. b) Use the graph to estimate the meeting time and meeting distance from Harbor A. c) Use the two functions to confirm the meeting time and distance exactly.

Four lines in the coordinate plane all pass through the same point \(P\). a) Find the coordinates of \(P\). b) Find the equation of each of the four lines. c) A fifth line \(h_5\) also passes through \(P\) and is parallel to the x-axis. Find its equation.

The point \(A(0, 1.5)\) is shown in the coordinate plane. a) Find the equations of the three lines \(g_1\), \(g_2\), and \(g_3\), which all pass through \(A\). b) What parameter do all three equations have in common when written in the form \(y = mx + b\)? c) Give an equation for another line \(g_4\) that also passes through \(A\) and has a negative slope whose absolute value is less than the absolute value of the slope of \(g_1\).

The graph shows a family of parallel lines. a) Find the slope \(m\) that all the lines have in common. b) Write equations for lines \(k\), \(l\), and \(n\). c) Describe how the equation changes when a line is shifted vertically while remaining parallel. Which parameter changes?

A candle is \(8\,\text{in.}\) tall when it is lit. Its height \(h\), in inches, decreases at a constant rate as it burns for \(t\) hours. a) Use the graph to write the linear function \(h(t)\). Pay attention to the scale on each axis. b) Find the candle's height after \(12\) hours and check the value on the graph.

A line \(g\) passes through \(P(-1, 4)\) and \(Q(2, -2)\), as shown in the graph. a) Find the equation of line \(g\). b) Use substitution to determine whether \(R(4, -6)\) also lies on the line.

Write equations for lines \(g_1\), \(g_2\), and \(g_3\). Then read the intersection point of \(g_1\) and \(g_2\) directly from the graph.

A climber assumes that air temperature decreases at a constant rate as elevation increases. The climber measures \(61\,\text{°F}\) at \(2000\,\text{ft}\) and \(47\,\text{°F}\) at \(6000\,\text{ft}\). a) Use a linear model to predict the temperature at a \(10{,}000\)-foot summit. b) A plane flies at \(39{,}000\,\text{ft}\). What temperature does the model predict at that elevation? c) Why is it problematic to use this simple linear model at extreme elevations?

A cylindrical container starts empty and has a base area of \(200\,\text{cm}^2\). It is filled at a constant rate of \(500\,\text{mL}\) per minute. a) How many centimeters per minute does the water height increase? b) Write a function for the water height \(h\), in centimeters, after \(t\) minutes. c) How would the graph change for a container with a base area of \(100\,\text{cm}^2\) filled at the same rate? Compare the slopes. d) Suppose a container consists of two stacked cylinders. The lower cylinder has base area \(200\,\text{cm}^2\) and height \(10\,\text{cm}\). The upper cylinder has base area \(100\,\text{cm}^2\) and height \(10\,\text{cm}\). Describe the water-height graph and what happens after \(4\) minutes.

A cylindrical candle is initially \(12.5\,\text{in.}\) tall. After it is lit, its height decreases at a constant rate of \(0.6\,\text{in.}\) per hour. a) Write a function for the candle's height \(h\), in inches, after \(t\) hours. b) How long will it take the candle to reach half its original height? Give the answer in hours and minutes. c) Suppose the candle were shaped like a pyramid, becoming narrower toward the top. Explain why its height would not decrease linearly if the same volume of wax burned each hour.

A rectangular water tank holds at most \(375\) gallons. It already contains \(50\) gallons when filling begins. An inlet adds \(30\) gallons in \(8\) minutes at a constant rate. a) Find the inflow rate in gallons per minute and write a function \(V(t)\) for the volume, in gallons, after \(t\) minutes. b) How long will it take to fill the tank completely? Give the time in hours, minutes, and seconds. c) Suppose a blockage cuts the inflow rate in half. Describe how the slope of the graph and the time needed to fill the tank would change.

Two candles are lit at the same time. Candle A is \(5\,\text{cm}\) tall and burns down at a constant rate of \(10\,\text{mm}\) per hour. Candle B is \(12\,\text{cm}\) tall and, each hour, loses an amount equal to \(10\%\) of its original height. Use linear models to determine when the candles would have the same height. Then decide whether that actually happens while the candles are burning.

At the beginning of an observation, a smartphone battery is at \(10\%\). It is connected to a fast charger and reaches \(90\%\) after \(40\) minutes. The charger is then removed, and a \(20\)-minute video call lowers the battery level to \(60\%\). a) Write a function for battery level \(y\), in percent, as a function of elapsed time \(x\), in minutes, for each phase. b) Compare the charging rate with the magnitude of the discharging rate. By how many percentage points per minute do they differ? c) At what elapsed times was the battery level exactly \(75\%\)? d) If the phone continued to discharge at the same rate after the call, how many additional minutes would it take to reach \(0\%\)?