Question 771·Hard·Math

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The scatterplot shown relates a day’s average outdoor temperature, $t$ (in degrees Fahrenheit), to the amount of natural gas a building used for heating that day, $g$ (in therms). A line of best fit is drawn, and its equation is shown on the graph:

g=-4t+240.

According to this model, what is the predicted average temperature when the predicted gas usage is half the predicted gas usage at $t=30$ ?

A line of best fit predicts an output from an input, even if a plotted dot sits nearby. If a question asks for the input that gives a fraction of a prediction, evaluate the given model first. Then scale the output and solve backward with a Desmos regression. Halve the predicted output, not the input.

Hints

Hint 1
A model prediction comes from the line’s equation, not from a nearby dot. Replace the temperature variable with $30$ in the given equation. What gas use does the model predict?
Hint 2
The word half applies to the gas use you found. Gas use is the model’s output, so divide it by $2$ rather than dividing the temperature by $2$ .
Hint 3
Now the gas use is known, but the input temperature is missing. Set the model equal to the halved gas use and solve for temperature. Which quantity should your answer measure?

Step-by-step

Approach 1: Find the prediction, then solve backward

Step 1
Find the gas use at 30 degrees
Use the given line, not a plotted dot: a line of best fit gives the model’s prediction. In Desmos, type $f(x)=-4x+240$ , using $x$ for the graph’s temperature $t$ , and then type $f(30)$ . Desmos returns $120$ , so the model predicts $120$ therms at $30$ degrees Fahrenheit.
Step 2
Halve the predicted gas use
The question says half the predicted gas usage, so halve the output $120$ , not the input $30$ . Type $f(30)/2$ in Desmos. It returns $60$ therms. That’s the target gas use, not yet the temperature.
Step 3
Find the temperature for that gas use
Now the temperature is the unknown input. Type $f(x_1)\sim60$ in Desmos. The subscript makes $x_1$ a value Desmos can find, and $\sim$ asks it to match the known output. Under PARAMETERS, Desmos shows $x_1=45$ . So the predicted average temperature is $45$ degrees Fahrenheit. Choice C.

Approach 2: Use the slope to track the change

Step 1
Find how much gas use must drop
The model predicts $120$ therms at $30$ degrees, and the target is $60$ therms. Find the drop in the predicted output: $120-60=60\text{ therms}$ .
Step 2
Convert the drop to a temperature rise
The slope $-4$ means each rise of $1$ degree Fahrenheit predicts $4$ fewer therms. Divide the needed drop by $4$ therms per degree: $\frac{60}{4}=15$ . So the temperature must rise by $15$ degrees Fahrenheit.
Step 3
Add the rise to the starting temperature
Start at the given $30$ degrees Fahrenheit and add the $15$ -degree rise: $30+15=45$ . The predicted average temperature is $45$ degrees Fahrenheit. Choice C.

Question 771·Hard·Math

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