The scatterplot shows the relationship between time, , and the mass of a substance remaining, . A curve representing an exponential model is shown. The model can be written as , where and are positive constants. Two points on the model curve are labeled on the graph. Which choice is closest to the value of ?

Solution available with step-by-step explanation

SAT Two-Variable Data Question 36 | Free SAT Prep | Aniko

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Question 36·Hard·Two-Variable Data: Models and Scatterplots

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The scatterplot shows the relationship between time, $x$ , and the mass of a substance remaining, $y$ . A curve representing an exponential model is shown. The model can be written as $y=a(b)^x$ , where $a$ and $b$ are positive constants.

Two points on the model curve are labeled on the graph.

Which choice is closest to the value of $b$ ?

For an exponential model $y=a(b)^x$ , two points are enough to find $b$ quickly: write $y_1=a(b)^{x_1}$ and $y_2=a(b)^{x_2}$ , then divide to get $\frac{y_2}{y_1}=b^{x_2-x_1}$ . Be careful to use the difference in $x$ as the exponent, and remember that a residual ratio like $\tfrac12$ may require taking a root (here, a cube root) to solve for $b$ .

Hints

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Use the labeled points

Read the coordinates of the two points that are explicitly labeled on the curve.

Set up two equations

Plug each labeled point $(x,y)$ into $y=a(b)^x$ to create two equations involving $a$ and $b$ .

Eliminate $a$ by dividing

Divide one equation by the other so that $a$ cancels, leaving an equation with a power of $b$ like $b^{x_2-x_1}$ .

Desmos Guide

Compute the ratio and exponent difference

From the labeled points, compute $\frac{120}{240}=0.5$ and $4-1=3$ .

Evaluate the cube root in Desmos

Type (.5)^(1/3) and read the decimal value Desmos shows.

Match to the closest option

Compare the value you computed to the answer choices and select the closest one.

Step-by-step Explanation

Write equations from the two labeled points

From the graph, the labeled points on the model curve are $(1,240)$ and $(4,120)$ .

Substitute into $y=a(b)^x$ :

$240=a(b)^1$
$120=a(b)^4$

Divide to eliminate $a$

Divide the second equation by the first:

\frac{120}{240}=\frac{a(b)^4}{a(b)^1}=b^{4-1}=b^3

So $\tfrac12=b^3$ .

Solve for $b$

Take the cube root of both sides:

b=\left(\frac12\right)^{1/3}

Using a calculator, $\left(\tfrac12\right)^{1/3}\approx 0.794$ , so the closest choice is 0.79.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation