Aimathic – Dependent and independent variables

Dependent and independent variables

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Difficulty: 1 – 5

5497056

A faucet fills a tub over time. Let $m$ be the number of minutes and $g$ be the gallons added. Which variable is independent, and which is dependent?

Hints

- Ask which quantity can be selected first. - Identify which quantity changes as the other one changes.

Solution

1. The number of minutes $m$ is the input that can be chosen, so it is independent. 2. The gallons added $g$ change in response to time, so they are dependent.

Answer

Independent variable: $m$ Dependent variable: $g$

5497116

In the ordered pair $(3, 15)$ for the relationship $c=5n$, $n$ is the number of notebooks and $c$ is the cost in dollars. Explain what each coordinate means and identify the independent variable.

Hints

- Match the coordinate order to the variable order in the relationship. - Use the context to decide which quantity determines the other.

Solution

1. The first coordinate gives $n=3$ notebooks. 2. The second coordinate gives $c=15$ dollars. 3. The notebook count $n$ is independent, and the cost $c$ is dependent.

Answer

The pair means $3$ notebooks cost $\$15$. The independent variable is $n$, and the dependent variable is $c$.

5497166

For a cube, $V=s^3$, where $s$ is side length and $V$ is volume. Explain why $V$ is the dependent variable.

Hints

- Ask which value must be known before the other can be calculated. - Identify which quantity is produced by the formula.

Solution

1. A value of $s$ determines one specific value of $V$ through $V=s^3$. 2. Therefore, $V$ depends on the chosen side length $s$.

Answer

$V$ is dependent because its value is determined by the chosen side length $s$.

5497066

A craft booth charges $\$2$ per bead strand. Let $s$ be the number of strands and $c$ be the total cost. Identify the variable roles and write the equation.

Hints

- Separate the chosen count from the amount that results. - Use the per-item amount to connect the two variables.

Solution

1. The number of strands $s$ is chosen, so it is independent. 2. The total cost $c$ depends on that choice, so it is dependent. 3. The equation is $c=2s$.

Answer

Independent: $s$ Dependent: $c$ Equation: $c=2s$

5497076

A reading plan starts with $12$ pages already completed and adds $8$ pages each day. Let $d$ be days and $p$ be total pages completed. Which variable depends on the other, and what is the equation?

Hints

- Identify the starting amount separately from the repeated change. - Write the total number of pages in terms of the elapsed days.

Solution

1. The number of days $d$ is the independent variable. 2. The total pages $p$ depend on the days. 3. Starting at $12$ and adding $8$ per day gives $p=12+8d$.

Answer

Independent: $d$ Dependent: $p$ Equation: $p=12+8d$

5497086

For the equation $y=5x$, suppose $x$ is the number of bags and $y$ is the number of apples. Explain which variable is independent and what the coefficient $5$ means.

Hints

- Use the context to decide which quantity is counted first. - Interpret the numerical factor as a change in the dependent quantity for one unit of the independent quantity.

Solution

1. The number of bags $x$ is selected, so it is independent. 2. The number of apples $y$ depends on the bag count. 3. The coefficient $5$ means there are $5$ apples per bag.

Answer

Independent variable: $x$ Dependent variable: $y$ The $5$ means $5$ apples per bag.

5497096

A candle is $18$ centimeters tall and burns $2$ centimeters each hour. Let $h$ be hours and $r$ be the remaining height. Identify the variables and write an equation.

Hints

- Determine which quantity progresses and which quantity responds. - Use subtraction because the output decreases from a starting amount.

Solution

1. Time $h$ is independent. 2. Remaining height $r$ depends on time. 3. The candle starts at $18$ and decreases by $2$ per hour, so $r=18-2h$.

Answer

Independent: $h$ Dependent: $r$ Equation: $r=18-2h$

5497106

The table records a relationship. <table> <thead> <tr> <th>Number of trays $t$</th> <th>Number of cups $c$</th> </tr> </thead> <tbody> <tr> <td>$1$</td> <td>$6$</td> </tr> <tr> <td>$2$</td> <td>$12$</td> </tr> <tr> <td>$4$</td> <td>$24$</td> </tr> </tbody> </table> Which variable is independent, which is dependent, and what equation fits the table?

Hints

- Look at which column lists the chosen counts and which lists resulting totals. - Compare each output with its input to find the consistent relationship.

Solution

1. The tray count $t$ is the input and is independent. 2. The cup count $c$ changes with the tray count and is dependent. 3. Each tray corresponds to $6$ cups, so $c=6t$.

Answer

Independent: $t$ Dependent: $c$ Equation: $c=6t$

5497136

A recipe uses $\frac{3}{4}$ cup of oats per batch. Let $b$ be batches and $o$ be cups of oats. Write $o$ in terms of $b$ and identify both variable roles.

Hints

- Use the number of batches as the repeated-group count. - Multiply the amount per batch by the input count.

Solution

1. The batch count $b$ is independent. 2. The oat amount $o$ depends on the number of batches. 3. The equation is $o=\frac{3}{4}b$.

Answer

Independent: $b$ Dependent: $o$ Equation: $o=\frac{3}{4}b$

5497146

A parking meter starts with $30$ minutes and adds $15$ minutes for each quarter inserted. Let $q$ be quarters and $m$ be total minutes. Write the equation and identify the dependent variable.

Hints

- Separate the initial amount from the amount added repeatedly. - The output is the quantity that changes after the input count is chosen.

Solution

1. The quarter count $q$ is independent. 2. Total minutes $m$ depend on the number of quarters. 3. The equation is $m=30+15q$.

Answer

Equation: $m=30+15q$ Dependent variable: $m$

5497156

A square has side length $s$ and area $A$. State which variable is independent and write the equation relating them.

Hints

- The geometric measurement chosen first determines the resulting area. - Use the standard area relationship for a square.

Solution

1. The side length $s$ is selected, so it is independent. 2. The area $A$ depends on the side length. 3. The equation is $A=s^2$.

Answer

Independent: $s$ Dependent: $A$ Equation: $A=s^2$

5497186

A roll begins with $50$ tickets. After $s$ tickets are sold, $r$ tickets remain. Write an equation and identify the dependent variable.

Hints

- Start from the original total and represent what is removed. - The remaining amount changes in response to the sold amount.

Solution

1. The sold count $s$ is treated as independent. 2. The remaining count $r$ depends on how many are sold. 3. The equation is $r=50-s$.

Answer

Equation: $r=50-s$ Dependent variable: $r$

5497196

A science class cools a sample by $1.5$ degrees each minute from an initial $24$ degrees Celsius. Let $m$ be minutes and $T$ be temperature. Write the equation and identify the independent variable.

Hints

- Identify the initial measurement and the repeated change. - Elapsed time is the input that determines the later measurement.

Solution

1. Time $m$ is independent. 2. Temperature $T$ depends on elapsed time. 3. The sample starts at $24$ and decreases by $1.5$ per minute, so $T=24-1.5m$.

Answer

Equation: $T=24-1.5m$ Independent variable: $m$

5497216

A train moves at $45$ miles per hour. Choose variables, define them, and write an equation that gives distance in terms of time. Identify the variable roles.

Hints

- Choose variable names that match the two quantities. - Write the output quantity as the rate times the input quantity.

Solution

1. Let $t$ be time in hours and $d$ be distance in miles. 2. Time $t$ is independent, and distance $d$ is dependent. 3. At $45$ miles per hour, $d=45t$.

Answer

Let $t$ be time in hours and $d$ be distance in miles. Independent: $t$ Dependent: $d$ Equation: $d=45t$

5497236

A photo printer takes $4$ seconds per photo. Which equation correctly gives total time $t$ in terms of photo count $p$: $p=4t$, $t=4p$, or $t=p+4$? Identify the variable roles.

Hints

- Test each equation using one photo. - The dependent quantity should equal the per-item amount times the count.

Solution

1. The photo count $p$ is selected, so it is independent. 2. Total time $t$ depends on that count. 3. Four seconds for each photo gives $t=4p$.

Answer

Correct equation: $t=4p$ Independent: $p$ Dependent: $t$

5497276

A delivery drone’s height $H$ after $t$ seconds is modeled by $H=2t+6$. Identify the starting height, the independent variable, and the dependent variable.

Hints

- Look at the term that remains when the input is zero. - Use the context to identify which measurement changes as time passes.

Solution

1. At $t=0$, $H=6$, so the starting height is $6$. 2. Time $t$ is independent. 3. Height $H$ depends on time and is dependent.

Answer

Starting height: $6$ Independent variable: $t$ Dependent variable: $H$

5497286

A rectangular garden has a fixed width of $4$ feet and a variable length $l$. Let $A$ be area. Write the relationship and identify the dependent quantity.

Hints

- Use the fixed dimension with the variable dimension in the area relationship. - The measurement calculated from the chosen length is the dependent quantity.

Solution

1. The length $l$ is the independent variable. 2. Area is width times length. 3. With width $4$, $A=4l$. 4. The area $A$ is dependent.

Answer

Equation: $A=4l$ Dependent variable: $A$

5497296

In each situation, identify the dependent variable. Use the situation to decide which quantity changes as a result of the other; do not use equation-side position as a rule. a) A store chooses the number of pencil packs $x$ to order, and the total number of pencils $y$ is described by $7x=y$. b) A game charges a fixed fee plus an amount for each round. The number of rounds $n$ is chosen first, and the total cost $C$ satisfies $C=10+3n$. c) Maya starts with $\$40$. The amount she spends $s$ is chosen, and the amount remaining $r$ satisfies $40-r=s$.

Hints

- For each situation, identify which quantity is chosen or controlled first. - The dependent variable is the quantity whose value follows from that choice. - Check the context rather than relying on which side of the equals sign contains a variable.

Solution

1. In a), the number of ordered packs $x$ determines the total number of pencils, so $y$ is dependent. 2. In b), the chosen number of rounds $n$ determines the total cost, so $C$ is dependent. 3. In c), the chosen amount spent $s$ determines the amount remaining, so $r$ is dependent. 4. The dependent variable appears on different sides or in different positions in the three equations, so equation position is not the deciding feature.

Answer

a) $y$ b) $C$ c) $r$

5497306

A student can choose the number $k$ of laps to swim. Total distance $d$ is then calculated. Which variable belongs on the x-axis of a graph, and why?

Hints

- Identify the input quantity before thinking about axis placement. - Use the standard convention for placing the independent variable on a graph.

Solution

1. The lap count $k$ is chosen and is independent. 2. The independent variable is conventionally placed on the x-axis. 3. Therefore, $k$ belongs on the x-axis.

Answer

$k$ belongs on the x-axis because it is the independent variable.

5497396

A classroom has $5$ tables with $c$ chairs at each table, so total chairs are $T=5c$. Which quantity may vary in this equation, and which variable is dependent?

Hints

- Distinguish the fixed number from the quantities represented by variables. - The total changes when the per-table amount changes.

Solution

1. The number of tables is fixed at $5$. 2. Chairs per table $c$ may vary and is the independent variable. 3. Total chairs $T$ depend on $c$, so $T$ is dependent.

Answer

The varying independent quantity is $c$, chairs per table. The dependent variable is $T$, total chairs.

5497406

A graphing program asks for the independent variable first. For the relationship $M=7+0.5d$, where $d$ is days and $M$ is plant mass in grams, which variable should be entered first and what does $7$ represent?

Hints

- Identify the input from the context before using the equation. - Interpret the constant by considering a zero input.

Solution

1. Days $d$ are the independent variable, so $d$ should be entered first. 2. Mass $M$ is dependent. 3. At $d=0$, $M=7$, so $7$ is the initial mass in grams.

Answer

Enter $d$ first. The $7$ represents the initial mass of $7\,\text{g}$.

5107266

A snack mix contains $10$ scoops in all. Let $x$ be the number of fruit scoops and $y$ be the number of cereal scoops, so $x+y=10$. a) Give three ordered pairs of positive whole numbers $(x, y)$ that satisfy the equation. b) If $x$ is chosen first, explain why $y$ is the dependent variable.

Hints

- Use the fixed total to check each ordered pair. - Keep both coordinates positive whole numbers. - Ask which value is determined after the other is chosen.

Solution

1. Positive whole-number pairs whose coordinates add to $10$ include $(1, 9)$, $(4, 6)$, and $(7, 3)$. 2. Once $x$ is chosen, $y$ must equal $10-x$. Therefore, the value of $y$ depends on the chosen value of $x$.

Answer

a) Sample pairs: $(1, 9)$, $(4, 6)$, and $(7, 3)$ b) $y$ is dependent because $y=10-x$ once $x$ is chosen.

5107656

A teacher divides $24$ markers equally among $x$ groups. Let $T$ be the number of markers in each group, so $T=24\div x$. a) Find $T$ when $x=2$, $x=4$, and $x=8$. b) Identify the independent and dependent variables. c) Describe how $T$ changes for these values as $x$ increases.

Hints

- Substitute each number of groups into $T=24\div x$. - Decide which quantity is chosen and which results from that choice. - Compare the three outputs in order.

Solution

1. When $x=2$, $T=24\div2=12$. When $x=4$, $T=24\div4=6$. When $x=8$, $T=24\div8=3$. 2. The number of groups $x$ is independent because it is chosen first. The number in each group $T$ depends on $x$. 3. For the given values, increasing the number of groups decreases the number of markers in each group.

Answer

a) $T=12$, $6$, and $3$, respectively b) Independent: $x$; dependent: $T$ c) $T$ decreases as $x$ increases for the given values.

5203916

Maya makes a chain of triangles with craft sticks. The first triangle needs $3$ sticks. Each additional triangle needs only $2$ more sticks because neighboring triangles share one side. a) How many sticks are needed for a chain of $25$ triangles? b) Maya uses all $61$ sticks in a box. How many triangles are in the chain? c) Let $n$ be the number of triangles and $s$ be the number of sticks. Write an equation that relates $s$ to $n$.

Hints

- Compare what changes when one more triangle is added to the chain. - For the reverse question, undo the same relationship that connects the number of triangles to the number of sticks. - In the equation, make the number of sticks depend on the number of triangles.

Solution

a) The first triangle uses $3$ sticks and the other $24$ triangles add $2$ sticks each, so $3 + 24 \times 2 = 51$ sticks. b) The relationship can be reversed: $(61 - 1) \div 2 = 30$, so the chain has $30$ triangles. c) Each triangle contributes two sticks to the repeating pattern, with one additional stick at the end. The equation is $s = 2n + 1$.

Answer

a) $51$ sticks b) $30$ triangles c) $s = 2n + 1$

5226506

Consider the expression $12.5-x$. a) Evaluate the expression for $x=2$, $x=5$, and $x=10$. b) How does the value of the expression change as $x$ increases through these values? Explain.

Hints

- Substitute one value of $x$ at a time. - Compare the three results in the same order as the inputs. - Think about subtracting increasingly large amounts from a fixed number.

Solution

1. For $x=2$, $12.5-2=10.5$. 2. For $x=5$, $12.5-5=7.5$. 3. For $x=10$, $12.5-10=2.5$. 4. As $x$ increases from $2$ to $5$ to $10$, the value decreases from $10.5$ to $7.5$ to $2.5$. 5. A greater amount is being subtracted from the same starting value, so the result becomes smaller.

Answer

a) For $x=2$, the value is $10.5$; for $x=5$, it is $7.5$; for $x=10$, it is $2.5$. b) The value decreases as $x$ increases.

5497126

A student writes $d=4t$ for distance $d$ traveled in $t$ hours. Another student says $t$ must be dependent because it is on the right side of the equation. Evaluate that claim.

Hints

- Use the causal or input-output relationship in the context. - Do not decide variable roles from left-side and right-side placement alone.

Solution

1. Variable roles come from the situation, not merely from equation position. 2. Elapsed time $t$ is treated as the input. 3. Distance $d$ depends on time, so $t$ is independent and $d$ is dependent.

Answer

The claim is incorrect. $t$ is independent and $d$ is dependent.

5497176

A bicycle rack holds $2$ wheels for each bicycle. Let $b$ be bicycles and $w$ be wheels. A student writes $b=2w$. Correct the equation and identify the variable roles.

Hints

- Check the equation with one bicycle as a simple test case. - Place the resulting total in terms of the chosen count.

Solution

1. The bicycle count $b$ is independent. 2. The wheel count $w$ is twice the bicycle count. 3. The correct equation is $w=2b$. 4. The student reversed which quantity is multiplied by $2$.

Answer

Independent: $b$ Dependent: $w$ Correct equation: $w=2b$

5497206

The variable-role headings in the table are incorrect. <table> <thead> <tr> <th>Independent: total cost $c$</th> <th>Dependent: number of markers $n$</th> </tr> </thead> <tbody> <tr> <td>$3$</td> <td>$1$</td> </tr> <tr> <td>$6$</td> <td>$2$</td> </tr> <tr> <td>$9$</td> <td>$3$</td> </tr> </tbody> </table> Markers cost $\$3$ each. Correct the variable-role headings.

Hints

- Use the pricing situation rather than trusting the printed headings. - Ask which quantity can be selected before calculating the other.

Solution

1. The number of markers is chosen first. 2. The total cost changes according to that count. 3. Therefore, $n$ is independent and $c$ is dependent.

Answer

Independent variable: number of markers $n$ Dependent variable: total cost $c$

5497226

A school store’s total price $P$ for $n$ folders is $P=1.25n$. What changes when $n$ increases by $1$, and which variable is responsible for that change?

Hints

- Compare the equation for two consecutive input values. - Connect the coefficient to the change in the output for one added item.

Solution

1. The input $n$ increases by $1$. 2. The equation adds another $1.25$ to $P$. 3. Thus the dependent variable $P$ increases by $\$1.25$, caused by the change in independent variable $n$.

Answer

When $n$ increases by $1$, $P$ increases by $\$1.25$. The independent variable $n$ drives the change.

5497246

A tank contains $80$ liters and drains $5$ liters each minute. Complete the statement: “The amount remaining depends on ______ because ______.” Then write the equation using $m$ for minutes and $L$ for liters remaining.

Hints

- Explain the direction of dependence in words before writing symbols. - Represent the fixed start and the repeated loss separately.

Solution

1. The amount remaining changes according to elapsed time. 2. Therefore, liters remaining depend on minutes because time determines how much has drained. 3. The equation is $L=80-5m$.

Answer

The amount remaining depends on minutes because elapsed time determines how much has drained. $L=80-5m$

5497256

For $P=4s$, where $s$ is the side length of a square and $P$ is its perimeter, compare what happens to $P$ when $s$ changes from $3$ to $5$. Which variable was changed directly?

Hints

- Evaluate the relationship at both input values. - Distinguish the value selected directly from the value recalculated afterward.

Solution

1. At $s=3$, $P=4\times3=12$. 2. At $s=5$, $P=4\times5=20$. 3. The perimeter increases by $8$. 4. The side length $s$, the independent variable, was changed directly.

Answer

$P$ changes from $12$ to $20$, an increase of $8$. The directly changed variable is $s$.

5497266

A table lists hours practiced and songs learned. <table> <thead> <tr> <th>Hours $h$</th> <th>Songs $s$</th> </tr> </thead> <tbody> <tr> <td>$1$</td> <td>$2$</td> </tr> <tr> <td>$2$</td> <td>$4$</td> </tr> <tr> <td>$3$</td> <td>$6$</td> </tr> </tbody> </table> A student claims songs should be independent because songs appear in the right column. Explain the error.

Hints

- Use the meaning of the quantities rather than their display order. - Ask which quantity is treated as the input in the situation.

Solution

1. Table position does not determine variable roles. 2. In this model, practice time $h$ is the input. 3. Songs learned $s$ depend on the hours practiced. 4. Therefore, $h$ is independent and $s$ is dependent.

Answer

The claim is incorrect. Hours $h$ are independent, and songs $s$ are dependent; column position alone does not decide the roles.

5497316

A weather station records time $t$ and air temperature $T$. Can the station choose the temperature first and make time depend on it in this data collection? Explain the intended variable roles.

Hints

- Use the design of the data collection to identify the input. - Variable roles can depend on how a relationship is being studied.

Solution

1. In this observation plan, time is the scheduled input for each measurement. 2. The recorded temperature is observed at each time. 3. Therefore, $t$ is independent and $T$ is dependent in the data relationship.

Answer

In this data collection, $t$ is independent and $T$ is dependent because temperature is recorded at selected times.

5497326

A ferry charges $\$6$ to board plus $\$2$ per bicycle. Let $b$ be bicycles and $C$ be total cost. Write the equation, identify the variables, and state the meaning of $C$ when $b=0$.

Hints

- Separate the fixed amount from the amount tied to the input count. - Interpret the output when the independent variable is zero.

Solution

1. Bicycle count $b$ is independent, and cost $C$ is dependent. 2. The equation is $C=6+2b$. 3. At $b=0$, $C=\$6$, representing the boarding charge with no bicycles.

Answer

Independent: $b$ Dependent: $C$ Equation: $C=6+2b$ $C=\$6$ when $b=0$, the boarding charge alone.

5497336

A craft-stick display uses $6$ sticks for the first section and $4$ additional sticks for each later section. Let $n$ be the number of sections and $s$ be the number of sticks. Which variable is independent, and which equation fits: $s=4n+2$ or $s=6n+4$? Justify the equation from the pattern description.

Hints

- Separate the sticks for the first section from the sticks added by the remaining sections. - The number of later sections is one less than the total number of sections.

Solution

1. The number of sections $n$ is independent, and the number of sticks $s$ is dependent. 2. The first section uses $6$ sticks, and the remaining $n-1$ sections add $4$ sticks each. 3. Thus $s=6+4(n-1)=6+4n-4=4n+2$. 4. Therefore, the matching equation is $s=4n+2$.

Answer

Independent: $n$ Dependent: $s$ Equation: $s=4n+2$, because $6+4(n-1)=4n+2$.

5497346

The equation $a=12-p$ relates available seats $a$ to occupied seats $p$ in a $12$-seat van. If $p$ is chosen as the independent variable, explain how $a$ changes when $p$ increases by $1$.

Hints

- Compare the output before and after increasing the input by one. - The subtraction sign indicates the two quantities change in opposite directions.

Solution

1. The variable $a$ is dependent on $p$. 2. Replacing $p$ by $p+1$ subtracts one additional seat from $12$. 3. Therefore, $a$ decreases by $1$ when $p$ increases by $1$.

Answer

The dependent variable $a$ decreases by $1$ for each increase of $1$ in $p$.

5497356

A baker records oven time $t$ and number of loaves $L$ baked. The equation $L=3t$ is proposed, with $t$ in hours. What assumption about the variable relationship does this equation make? Identify the variable roles.

Hints

- Interpret what one unit of the input does to the output. - Notice what the equation predicts when the input is zero.

Solution

1. Time $t$ is independent, and loaves $L$ are dependent. 2. The coefficient $3$ means the model assumes $3$ loaves are baked per hour. 3. The equation also assumes the relationship continues at that constant rate from $0$.

Answer

Independent: $t$ Dependent: $L$ The equation assumes a constant rate of $3$ loaves per hour starting from $0$.

5497366

A table for $y=2x+1$ lists the pair $(4, 9)$. If $x$ is independent, explain why writing the pair as $(9, 4)$ changes its meaning.

Hints

- Use the input-output order of coordinates. - Check the reversed pair in the equation to see whether it still fits.

Solution

1. Ordered pairs list the independent value first and dependent value second. 2. The pair $(4, 9)$ means $x=4$ gives $y=9$. 3. The reversed pair $(9, 4)$ would claim $x=9$ gives $y=4$, which does not satisfy the equation.

Answer

$(4, 9)$ means input $4$ gives output $9$. Reversing it swaps the variable roles and produces a point that does not satisfy $y=2x+1$.

5497376

A concert hall adds rows while keeping $24$ seats in each row. Let $r$ be rows and $s$ be seats. State the domain type that makes sense for the independent variable and write the equation.

Hints

- Consider whether partial units of the independent object make sense. - Use the fixed number per group to form the output equation.

Solution

1. Row count $r$ is independent. 2. A row count must be a nonnegative whole number in this context. 3. Total seats $s$ depend on rows by $s=24r$.

Answer

Independent variable: $r$, with nonnegative whole-number values Dependent variable: $s$ Equation: $s=24r$

5497416

A student says an independent variable never changes. Correct the statement using the relationship $C=3n$, where $n$ is item count and $C$ is cost.

Hints

- Distinguish “independent” from “fixed.” - Use several possible item counts to see how the input can vary.

Solution

1. An independent variable may take different chosen values. 2. In $C=3n$, the item count $n$ can change from one case to another. 3. “Independent” means it is treated as the input, not that it is constant. 4. Cost $C$ depends on the chosen value of $n$.

Answer

The statement is false. An independent variable can change; it is the input whose chosen values determine the dependent variable.

5203906

Lucas builds staircases from cubes. Each step is two cubes wide. He records his observations in a table: <table> <tr><td>Staircase height</td><td>$1$</td><td>$2$</td><td>$3$</td></tr> <tr><td>Cubes in the bottom row</td><td>$2$</td><td>$4$</td><td>$6$</td></tr> <tr><td>Total number of cubes</td><td>$2$</td><td>$6$</td><td>$12$</td></tr> </table> a) For a staircase with height $10$, how many cubes are in the bottom row, and how many cubes are needed altogether? b) Let $h$ be the staircase height, $b$ the number of cubes in the bottom row, and $t$ the total number of cubes. Write equations that express $b$ and $t$ in terms of $h$.

Hints

- Compare the staircase height with the bottom-row count in each table column. - For the total, examine how the accumulated number of cubes changes as the height increases. - In part b), write each changing cube count as a dependent quantity determined by $h$.

Solution

a) The bottom row has twice as many cubes as the height, so $b = 2 \times 10 = 20$. The row counts are $2, 4, 6, \ldots, 20$, whose total is $110$ cubes. b) The bottom-row relationship is $b = 2h$. The total number of cubes is the sum of the first $h$ even numbers, which is $t = h(h + 1)$. For $h = 10$, this gives $t = 10 \times 11 = 110$.

Answer

a) Bottom row: $20$ cubes; total: $110$ cubes b) $b = 2h$ and $t = h(h + 1)$

5203926

A square flower bed is surrounded by one row of square paving stones. The side length of the bed is measured in stone lengths. <table> <tr><td>Side length of flower bed</td><td>$1$</td><td>$2$</td><td>$3$</td></tr> <tr><td>Number of paving stones</td><td>$8$</td><td>$12$</td><td>$16$</td></tr> </table> a) How many paving stones are needed when the side length is $12$? b) Let $n$ be the side length and $p$ be the number of paving stones. Write an equation that relates $p$ to $n$.

Hints

- Compare consecutive rows of the table to identify how the paving-stone count changes. - Relate the changing paving-stone count to the changing side length. - In part b), write the paving-stone count as the dependent quantity.

Solution

a) The number of paving stones increases by $4$ whenever the side length increases by $1$. For side length $12$, $4 \times 12 + 4 = 52$, so $52$ paving stones are needed. b) Four side lengths contribute $4n$, with $4$ additional corner stones. The equation is $p = 4n + 4$.

Answer

a) $52$ paving stones b) $p = 4n + 4$

5497386

A runner’s distance $d$ is given by $d=6t$. A student says doubling $d$ will always cause $t$ to double, so $d$ should be independent. Explain why that conclusion about variable roles does not follow.

Hints

- Separate a numerical pattern from the definition of input and output in the model. - Use the stated way the relationship is generated to assign roles.

Solution

1. The equation does imply corresponding doubled values in this relationship. 2. However, variable roles are set by the model: time $t$ is chosen or observed as the input. 3. Distance $d$ is calculated from time and remains the dependent variable.

Answer

The doubling relationship does not determine the roles. In this model, $t$ is the independent input and $d$ is the dependent output.

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