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Difficulty: MediumLinear Functions and Graphs

A scientist is measuring the temperature of a chemical compound during an experiment. The temperature of the compound decreases at a constant rate. At t=4t = 4 minutes after the experiment begins, the temperature of the compound is 72∘C72^\circ\text{C}. At t=12t = 12 minutes, the temperature of the compound is 48∘C48^\circ\text{C}. Which of the following functions TT models the temperature, in degrees Celsius, of the compound tt minutes after the experiment begins, where 0≤t≤250 \leq t \leq 25?

  1. T(t)=−3t+84T(t) = -3t + 84Answer
  2. B
    T(t)=−3t+72T(t) = -3t + 72
  3. C
    T(t)=−13t+2203T(t) = -\frac{1}{3}t + \frac{220}{3}
  4. D
    T(t)=−3t+60T(t) = -3t + 60

Answer

T(t)=−3t+84T(t) = -3t + 84
To find the linear model T(t)=mt+bT(t) = mt + b, we first calculate the slope mm using the given points (4,72)(4, 72) and (12,48)(12, 48). The slope is the change in temperature divided by the change in time: m=48−7212−4=−3m = \frac{48 - 72}{12 - 4} = -3. Next, we find the y-intercept bb by substituting one of the points into the equation T(t)=mt+bT(t) = mt + b. Substituting t=4t = 4 and T(4)=72T(4) = 72 yields 72=−3(4)+b72 = -3(4) + b, which simplifies to 72=−12+b72 = -12 + b. Adding 1212 to both sides gives b=84b = 84. Therefore, the correct linear function modeling the temperature is T(t)=−3t+84T(t) = -3t + 84.

Step-by-Step Solution

1
Calculate the rate of temperature change (slope) per minute.
The slope mm is −3-3.
Since the temperature decreases at a constant rate, the relationship is linear. The slope mm is the change in temperature divided by the change in time: m=48−7212−4=−248=−3m = \frac{48 - 72}{12 - 4} = \frac{-24}{8} = -3.
2
Find the initial temperature (y-intercept) at t=0t = 0.
The y-intercept bb is 8484.
Substitute the point (4,72)(4, 72) and the slope m=−3m = -3 into the slope-intercept equation T(t)=mt+bT(t) = mt + b. This gives 72=−3(4)+b72 = -3(4) + b, which simplifies to 72=−12+b72 = -12 + b. Adding 1212 to both sides yields b=84b = 84.
3
Write the final equation.
T(t)=−3t+84T(t) = -3t + 84
Combine the slope of −3-3 and the y-intercept of 8484 to complete the linear model equation.

Key Concept

Writing and interpreting linear functions from two points

Alternative Method

Instead of deriving the function algebraically, substitute the given values of t=4t = 4 and t=12t = 12 into each of the given options. Only the correct option yields the corresponding temperatures of 72 and 48 degrees Celsius, respectively.
Estimated Time:1m 30s
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Topics

Linear Functions and Graphs

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