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Interpret slope and intercept in context

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5512769
A bike-rental shop models the total cost by \(C(h)=8h+15\), where \(h\) is the number of rental hours and \(C(h)\) is the cost in dollars. Interpret the slope and the y-intercept in this situation. Include units.

Hints

- Connect the coefficient of \(h\) with how the cost changes when rental time increases. - Evaluate what the model says when the number of rental hours is \(0\). - Match each numerical parameter with the units it has in context.

Solution

1. The slope is \(8\). This means the cost increases by \(\$8\) for each additional hour of rental time. 2. The y-intercept is \(15\). Since \(C(0)=15\), it represents a fixed \(\$15\) charge before any rental hours are added.

Answer

The slope means \(\$8\) per rental hour. The y-intercept means there is a fixed \(\$15\) charge at \(0\) hours.
5290379
A market analyst models the supply price for a product by \(p(x)=1.5x+20\), where \(x\) is the number of units and \(p(x)\) is the price per unit in dollars. A \(\$5\) subsidy per unit is paid directly to producers, so producers can request \(\$5\) less from buyers for the same total payment. a) Write the new supply-price function \(p_{\mathrm{sub}}(x)\). b) Explain why the subsidy is represented by a downward vertical shift. c) Find and interpret the new y-intercept within this model.

Hints

- Determine how much buyers must pay after the producer receives the subsidy. - Subtracting the same constant from every output creates a vertical shift. - Evaluate the new function at \(x=0\). - Interpret the intercept only within the stated model.

Solution

1. Subtract the subsidy from every modeled price: \(p_{\mathrm{sub}}(x)=p(x)-5=1.5x+15\). 2. Because every output decreases by the same amount, every point on the graph moves down \(5\) units. 3. At \(x=0\), \(p_{\mathrm{sub}}(0)=15\), so the y-intercept is \((0, 15)\). 4. Within the algebraic model, the intercept means the modeled price is \(\$15\) when the quantity is \(0\). It should not automatically be interpreted as a real-world minimum operating price without additional economic assumptions.

Answer

a) \(p_{\mathrm{sub}}(x)=1.5x+15\) b) The subsidy subtracts \(5\) from every output, so the graph shifts down \(5\) units. c) The y-intercept is \((0, 15)\), representing a modeled price of \(\$15\) at a quantity of \(0\).
5512789
A school club models its profit from selling event tickets by \(P(n)=12n-180\), where \(n\) is the number of tickets sold and \(P(n)\) is the profit in dollars. a) Interpret the slope. b) Interpret the y-intercept, including why a negative value is meaningful here. c) State a reasonable contextual domain for \(n\).

Hints

- Ask what one additional ticket changes in the model. - Evaluate the model conceptually at zero tickets and interpret the sign of the result. - Use the fact that \(n\) counts physical tickets when deciding which input values make sense.

Solution

1. The slope is \(12\), so each additional ticket sold increases profit by \(\$12\). 2. The y-intercept is \(-180\). At \(n=0\), the club has a profit of \(-\$180\), meaning it has \(\$180\) in upfront costs before any tickets are sold. 3. Ticket counts cannot be negative or fractional, so a reasonable domain is the nonnegative integers.

Answer

a) Profit increases by \(\$12\) for each ticket sold. b) The y-intercept \(-\$180\) represents \(\$180\) in upfront costs before any tickets are sold. c) \(n\) should be a nonnegative integer.
5288179
In a geothermal model, underground temperature increases approximately linearly with depth. The temperature is \(25\,\text{°C}\) at a depth of \(500\,\text{m}\) and \(70\,\text{°C}\) at a depth of \(2000\,\text{m}\). a) Write a linear function \(T(d)\) for temperature as a function of depth \(d\), measured in meters. b) What surface temperature does the model predict? c) At what depth does the model predict a temperature of \(100\,\text{°C}\)?

Hints

- Treat depth as the input and temperature as the output. - Use the two data points to find the rate of change. - The y-intercept represents the model's temperature at depth \(0\). - To find a depth from a temperature, set the function equal to that temperature and solve for \(d\).

Solution

1. Use the points \((500, 25)\) and \((2000, 70)\). The slope is \(m=\frac{70-25}{2000-500}=\frac{45}{1500}=0.03\) degrees Celsius per meter. 2. Substitute \((500, 25)\) into \(T(d)=0.03d+b\): \(25=0.03\cdot 500+b\), so \(b=10\). Therefore, \(T(d)=0.03d+10\). 3. At the surface, \(d=0\), so \(T(0)=10\). The predicted surface temperature is \(10\,\text{°C}\). 4. Set \(T(d)=100\): \(100=0.03d+10\). Thus, \(90=0.03d\), so \(d=3000\,\text{m}\).

Answer

a) \(T(d)=0.03d+10\) b) \(10\,\text{°C}\) c) \(3000\,\text{m}\)
5288279
An energy company offers two annual residential plans. The Flex plan has a base charge of \(\$120\) and an energy rate of \(6.0\) cents per kWh. The Active plan has a base charge of \(\$180\) and an energy rate of \(4.5\) cents per kWh. 1. Find the annual usage at which the two plans cost the same. 2. The company raises the Active plan's base charge by \(20\%\). Find the new break-even usage and state when Active becomes less expensive. 3. Explain how lowering the Active plan's energy rate would affect the intersection of the two cost graphs, assuming all other values remain unchanged.

Hints

- Convert cents per kWh to dollars per kWh before writing the functions. - Equal costs correspond to equal function values. - A percentage increase in a base charge changes the y-intercept. - A lower energy rate produces a smaller slope.

Solution

1. In dollars, the cost functions are \(F(x)=0.06x+120\) and \(A(x)=0.045x+180\). Set them equal: \(0.06x+120=0.045x+180\). Then \(0.015x=60\), so \(x=4000\). The plans cost the same at \(4000\,\text{kWh}\). 2. The new Active base charge is \(180\cdot 1.20=\$216\). Solve \(0.06x+120=0.045x+216\): \(0.015x=96\), so \(x=6400\). Active is less expensive for \(x>6400\,\text{kWh}\). 3. Lowering Active's energy rate decreases the slope of its cost graph. Because its base charge remains higher, the graphs intersect at a lower usage value, so Active becomes advantageous sooner.

Answer

1. \(4000\,\text{kWh}\) 2. New break-even usage: \(6400\,\text{kWh}\); Active costs less for \(x>6400\,\text{kWh}\). 3. The intersection moves left, to a lower usage value.
5288389
Two monthly phone plans are modeled by linear functions, where \(x\) is data usage in gigabytes and \(K(x)\) is the monthly cost in dollars. Plan A: \(K_A(x)=2x+20\) Plan B: \(K_B(x)=3x+5\) 1. Identify \(m\) and \(b\) for each plan and explain what each parameter means in context. 2. Determine when Plan A costs less than Plan B.

Hints

- In \(K(x)=mx+b\), identify the part that changes with usage and the fixed part. - Use the units to interpret the slope and intercept. - Write a strict inequality to represent “Plan A costs less.”

Solution

1. For Plan A, \(m=2\) and \(b=20\). The rate is \(\$2\) per gigabyte, and the monthly base charge is \(\$20\). 2. For Plan B, \(m=3\) and \(b=5\). The rate is \(\$3\) per gigabyte, and the monthly base charge is \(\$5\). 3. Compare the plans: \(2x+20<3x+5\). Subtracting \(2x\) and \(5\) gives \(15<x\). Therefore, Plan A costs less when usage exceeds \(15\) GB.

Answer

1. Plan A: \(m=2\) dollars per GB, \(b=\$20\). Plan B: \(m=3\) dollars per GB, \(b=\$5\). 2. Plan A costs less when \(x>15\) GB.
5512779
The graph shows the height \(h\), in centimeters, of a candle \(t\) hours after it is lit. The model is linear over the time shown. a) What does the y-intercept mean in this context? b) Find and interpret the slope, including units. c) According to the model, what is the candle's height after \(5\) hours?
Figure for problem 551277

Hints

- Read the graph where \(t=0\) to interpret the intercept. - Use two clear graph points and compare vertical change with horizontal change. - The sign of the slope should agree with whether the candle gets taller or shorter over time.

Solution

1. From the graph, the y-intercept is \(12\), so the candle is \(12\,\text{cm}\) tall when it is lit. 2. Using the points \((0,12)\) and \((2,9)\), the slope is \(\frac{9-12}{2-0}=-\frac{3}{2}=-1.5\). The candle's height decreases by \(1.5\,\text{cm}\) per hour. 3. Continuing the linear model to \(t=5\) gives \(h=12-1.5\cdot5=4.5\). The predicted height is \(4.5\,\text{cm}\).

Answer

a) The candle's initial height is \(12\,\text{cm}\). b) The slope is \(-1.5\,\text{cm per hour}\), meaning the candle loses \(1.5\,\text{cm}\) of height each hour. c) \(4.5\,\text{cm}\)
5512799
A gym charges a one-time signup fee of \(\$35\) plus the same membership charge each month. After \(6\) months, a member has paid a total of \(\$185\). a) Write a linear model \(C(m)\) for the total amount paid after \(m\) months. b) Interpret both parameters of your model in context. c) Another member says the same plan should cost \(\$285\) after \(10\) months. Is that consistent with your model? Justify your answer.

Hints

- Identify which given amount belongs to month \(0\). - The total increase after several months comes from repeated equal monthly charges. - After writing the model, test the final claim by evaluating it at the stated month.

Solution

1. The y-intercept is the signup fee, so \(C(0)=35\). 2. The monthly charge is \(\frac{185-35}{6}=25\) dollars per month. 3. The model is \(C(m)=25m+35\). The slope \(25\) is the monthly membership charge, and the y-intercept \(35\) is the one-time signup fee. 4. \(C(10)=25\cdot10+35=285\), so the second member's statement is consistent with the model.

Answer

a) \(C(m)=25m+35\) b) The slope is a \(\$25\)-per-month charge, and the y-intercept is the \(\$35\) signup fee. c) Yes. The model gives \(C(10)=\$285\).
5288289
An energy company models a family of annual plans by \(K_p(x)=px+(300-2000p)\), where \(x\) is annual usage in kWh, \(p\) is the price per kWh in dollars, and \(K_p(x)\) is annual cost in dollars. 1. For a plan with \(p=0.12\), identify the slope and y-intercept and interpret both in context. 2. Show that all graphs in the family pass through one common point. Find the point and interpret it in context. 3. A customer uses \(3500\,\text{kWh}\) per year. Determine whether a higher or lower value of \(p\) gives this customer a lower annual cost.

Hints

- Substitute the specified value of \(p\) before interpreting the slope and intercept. - Rewrite the family so the coefficient of \(p\) is easy to see. - Find the usage value that makes the \(p\)-dependent part equal to \(0\). - For the final comparison, substitute \(3500\) and see how the cost changes as \(p\) changes.

Solution

1. With \(p=0.12\), \(K_{0.12}(x)=0.12x+60\). The slope \(0.12\) means each additional kWh adds \(\$0.12\) to the modeled annual cost. The y-intercept \(60\) means the model gives an annual cost of \(\$60\) at zero usage. 2. Rewrite the family as \(K_p(x)=p(x-2000)+300\). At \(x=2000\), the term containing \(p\) is \(0\), so every plan gives \(K_p(2000)=300\). The common point is \((2000,300)\): every plan costs \(\$300\) at \(2000\,\text{kWh}\). 3. At \(x=3500\), \(K_p(3500)=300+1500p\). This increases as \(p\) increases, so a lower value of \(p\) gives the lower annual cost.

Answer

1. For \(p=0.12\), slope \(=0.12\) dollars per kWh and y-intercept \(=60\) dollars. The slope is the added cost per additional kWh; the intercept is the modeled cost at zero usage. 2. Common point: \((2000,300)\). At \(2000\,\text{kWh}\), every plan costs \(\$300\). 3. A lower value of \(p\) gives the lower annual cost at \(3500\,\text{kWh}\).

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