Question 140·Medium·Linear Functions
A candle is 20 centimeters tall at exactly 6:00 PM. It burns at a constant rate, and the height of the candle, , in centimeters, hours after 6:00 PM can be modeled by the equation
Which of the following best describes the meaning of the number in this equation?
For questions asking what a number in a linear model means, first label what each variable represents from the context. Then match the equation to : the constant term is the starting value, and the coefficient of the variable is the rate of change (slope), including its sign and units. Finally, scan the choices for the one that correctly describes this rate using the correct quantities and units (for example, “change in height per hour”), and eliminate any options that treat the number as a starting value, a total time, or that mix up which number is which.
Hints
Focus on what h and t represent
Look back at the description under the equation: what does stand for, and what does stand for in this situation with the candle?
Think about the general form of a line
Compare to the form . In that form, what does the coefficient usually represent?
Use the sign and units of −0.8
The number is negative and is multiplied by time in hours. Ask yourself: does that sound more like a starting amount, a total time, or a rate at which something is changing?
Eliminate mismatched units and meanings
Check each answer: is it describing a time, a starting height, or a change in height per hour? Only one should match what the coefficient of means.
Desmos Guide
Graph the candle’s height equation
In Desmos, type h = 20 - 0.8t. This will draw a straight line showing how height changes as time increases.
Use a table to see the pattern
Click on the gear icon next to the equation and choose "Convert to table" (or add a table with in the first column and 20 - 0.8t in the second). Fill in several values like 0, 1, 2, 3, etc., and watch how changes each time increases by 1 hour. The amount goes up or down each time is what the coefficient represents.
Relate the pattern to the context
Once you see how changes in the table for each 1-hour step in , think about what that repeated change means in words for the height of the candle as it burns.
Step-by-step Explanation
Understand the variables and the context
The equation is
You are told that:
- is the height of the candle in centimeters.
- is the number of hours after 6:00 PM.
So this equation describes how the candle’s height changes over time as it burns.
Identify what each number in the equation represents
Compare this to the general form of a linear equation:
Here:
- is like , the starting value when . That means at 6:00 PM, the candle is 20 cm tall.
- is like , the coefficient of . In a linear model, this coefficient tells you how much the output (here, height ) changes when the input (time ) increases by 1 unit (1 hour).
Interpret the sign and units of the coefficient
Because the coefficient of is negative ():
- The height is going down over time, not up.
- The units of this coefficient come from “change in height” divided by “change in time,” so it is measured in centimeters per hour.
So represents a rate of change: how many centimeters of height are lost for each hour that passes.
Match this interpretation to the answer choices
From the reasoning above, is the rate at which the candle’s height decreases, measured in centimeters per hour, as time increases.
The choice that correctly states this idea is: “The height of the candle decreases by 0.8 centimeter every hour it burns.”