Question 682·Medium·Algebra

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A nutrition researcher found that during a moderate indoor-cycling session, the number of calories burned, $y$ , can be modeled by the linear equation

y = 9.4x + 70,

where $x$ is the number of minutes spent cycling ( $5 \le x \le 40$ ) and $y$ is the total calories burned.

If a participant burned 235 calories during a session, which of the following is closest to the predicted number of minutes the participant spent cycling?

A linear model uses an input to predict an output. If the question gives the output but asks for the input, replace the output in the equation with the given value, then use a one-variable regression in Desmos to solve. Don't divide the total by the rate alone: that treats the model's fixed amount as if the rate produced it.

Hints

Hint 1
In a linear model, $x$ is the input and $y$ is the output. Here, the calories are given, but the minutes aren't. Which variable should you replace with $235$ ?
Hint 2
The fixed $70$ is part of the total calories, so keep it when you write the equation. After setting the model equal to $235$ , solve for the minutes rather than dividing $235$ by $9.4$ .

Step-by-step

Set the model equal to the given calories
Step 1
Turn the given calories into an equation
The model calls $y$ the total calories, and the participant burned 235 calories, so replace $y$ with $235$ :
$235=9.4x+70.$
Here $x$ still means minutes. The $70$ is a constant term: it stays in the model regardless of the value of $x$ . When an output is given, set the model equal to that output to find the input.
Step 2
Solve for minutes in Desmos
Type $9.4x_1+70\sim235$ . Change $x$ to $x_1$ and $=$ to $\sim$ so Desmos solves for a number instead of graphing an equation. The regression shows $x_1\approx17.553$ under PARAMETERS. That's the predicted time in minutes before choosing an option.
Step 3
Choose the closest time
The result $17.553$ is within the allowed range $5\le x\le40$ . Of the listed times, $18$ is closest to $17.553$ , so the predicted session lasted about $18$ minutes. Choice C.

Question 682·Medium·Algebra

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