Super-Hard SAT Data Analysis Question 19 | Free SAT Prep

00:00

Try Full-Screen!

Question 19·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

0 / 50 Questions completed

A snack shop recorded the daily high temperature, $x$ , in degrees Celsius and the number of smoothies sold, $y$ , in hundreds. For temperatures within the range of the data, the relationship is modeled by the line of best fit $y=0.75x-6$ .

The shop changes its records so that temperature is represented by $F$ , in degrees Fahrenheit, and smoothie sales are represented by $N$ , the number of individual smoothies. The relationships between the variables are $F=1.8x+32$ and $N=100y$ .

Which choice gives the equation of the same line of best fit in terms of $F$ and $N$ ?

When a line of best fit is expressed using new units, do not try to change only the slope or only the intercept. First write each original variable in terms of the new variables, substitute into the model, and then simplify. A conversion that includes an added constant, such as the Celsius-to-Fahrenheit conversion, changes the intercept as well as the slope.

Hints

Click me!

Undo the temperature conversion

Solve $F=1.8x+32$ for $x$ so the original line can be written using $F$ .

Use the new sales unit

The original output $y$ is in hundreds, while $N$ counts individual smoothies. Relate them using $N=100y$ .

Substitute before comparing

Replace $x$ in $y=0.75x-6$ , then multiply the resulting expression by $100$ to obtain an equation involving $N$ .

Desmos Guide

Use algebra first

Algebra is faster here because the task is to rewrite a line after changing both input and output units.

Graph the converted model

In Desmos, let the horizontal variable $x$ represent Fahrenheit and the vertical variable $y$ represent individual smoothies. Enter $y=100(0.75((x-32)/1.8)-6)$ .

Check the matching equation

Enter $y=(125x-5800)/3$ . The two graphs should lie exactly on top of each other, verifying that the equations represent the same line.

Step-by-step Explanation

Rewrite the Celsius variable

From $F=1.8x+32$ , solve for $x$ :

x=\frac{F-32}{1.8}

Convert the sales variable and substitute

Because $N=100y$ , multiply the original model by $100$ after substituting for $x$ :

\begin{aligned} N&=100\left(0.75\left(\frac{F-32}{1.8}\right)-6\right)\\ &=\frac{125}{3}(F-32)-600 \end{aligned}

Simplify the equation

Multiply by $3$ and distribute:

\begin{aligned} 3N&=125(F-32)-1800\\ &=125F-5800 \end{aligned}

This is choice A.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation