Question 28·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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The scatterplot shows data for several solar-panel installations. The axes use scaled units, and the line of best fit is labeled in the graph.

Let $p$ be the total number of solar panels in an installation, and let $E$ be the monthly energy generated, in kilowatt-hours. Which choice gives the equation of the line of best fit in terms of $p$ and $E$ ?

A line of best fit summarizes the pattern in scattered data. With scaled axes, its labeled equation uses graph units, which may differ from the totals the question asks for. Evaluate the line at a convenient input in Desmos, convert the resulting point, and test the choices. Convert both coordinates before comparing equations.

Hints

Hint 1
The horizontal input counts tens of panels. A graph input of $1$ represents $10$ actual panels. If you test an equation at $p=10$ , which input should you use in the labeled line?
Hint 2
The vertical unit is hundreds of kilowatt-hours. Once the line predicts a graph height, multiply it by $100$ to get the energy $E$ that appears in the answer choices.
Hint 3
An equation for the same line must give the same energy at the panel count you chose. Substitute that count into each choice; a different prediction rules the choice out.

Step-by-step

Approach 1: Convert a point and test the choices

Step 1
Choose a panel count
The horizontal axis says “Solar panels (tens),” so a graph input of $x=1$ represents $p=10$ actual panels. Use that count to test the choices.
Step 2
Translate the energy scale
The vertical axis says “Energy (hundreds kWh),” so a graph height $y$ represents $E=100y$ kilowatt-hours. $y$ is not yet the energy total used in the choices.
Step 3
Find the graph’s prediction
Enter the labeled line as a function: type $f(x)=1.8x+4.5$ , then $f(1)$ . Desmos shows $6.3$ . At $10$ panels, the line predicts a graph height of $6.3$ hundreds of kilowatt-hours.
Step 4
Convert that prediction to kilowatt-hours
Type $100f(1)$ . Desmos shows $630$ , so the predicted energy for $p=10$ is $E=630$ kilowatt-hours, not $6.3$ kilowatt-hours.
Step 5
Test the four equations
Type each choice’s right-hand side with $p=10$ . In order, Desmos shows $22.5$ , $2250$ , $630$ , and $184.5$ . Only $E=18p+450$ gives the required $630$ kilowatt-hours for $10$ panels. Choice C.

Approach 2: Rewrite the whole line

Step 1
Put the totals into the labeled equation
To get a rule for every panel count, use $x=\frac{p}{10}$ because the graph counts tens of panels, and $y=\frac{E}{100}$ because it counts hundreds of kilowatt-hours. Substitute both conversions into the labeled line: $\frac{E}{100}=1.8\left(\frac{p}{10}\right)+4.5$ .
Step 2
Change the output to kilowatt-hours
Multiply both sides by $100$ to put $E$ by itself: $E=100\left[1.8\left(\frac{p}{10}\right)+4.5\right]$ . The $100$ multiplies the whole prediction, including both terms.
Step 3
Find the new slope and intercept
Distribute the $100$ . Type $100(1.8)/10$ and $100(4.5)$ in Desmos; it shows $18$ and $450$ . So the line predicts energy in kilowatt-hours from the total number of panels: $E=18p+450$ . Choice C.

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Question 28·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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