Question 21·Hard·Two-Variable Data: Models and Scatterplots
The scatterplot shown relates a day’s average outdoor temperature, (in degrees Fahrenheit), to the amount of natural gas a building used for heating that day, (in therms). A line of best fit is drawn, and its equation is shown on the graph:
According to this model, what is the predicted average temperature when the predicted gas usage is half the predicted gas usage at ?
When a problem says an output is a fraction (like half) of the model’s output at a given input, do it in two phases: (1) evaluate the model at the given input to get the baseline predicted value, then (2) apply the fraction to set a new target output and solve the linear equation to find the corresponding input. Be careful with negative slopes and when moving terms across the equals sign.
Hints
Find the predicted gas usage at
Use the equation and substitute .
Take half of that prediction
Whatever value you got for , divide it by 2. That’s the new target gas usage.
Work backward to the temperature
Set equal to the half-value and solve for .
Desmos Guide
Define the model
Enter g(t) = -4t + 240.
Compute the half-usage target from
Enter g30 = g(30) and then half = g30/2.
Solve for the temperature that gives the half-usage
Enter -4t + 240 = half and use Desmos to find the solution for .
Step-by-step Explanation
Predict gas usage at
Substitute into :
Halve the predicted usage
Compute the predicted usage from the previous step, then take half of it. Call this half-value .
This is the gas usage the question says to match.
Set up an equation to find the temperature
Set the model equal to and solve for :
Compute and select the matching choice
From Steps 1–2, , so .
Then
So the predicted average temperature is 45 degrees Fahrenheit.