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Difficulty: MediumInterpreting Linear Relationships in Context

A company uses a fleet of refrigerated delivery trucks to transport fresh produce. The temperature inside a truck's cooling compartment, TT, in degrees Fahrenheit (∘F^\circ\text{F}), tt minutes after the cooling system is turned on is modeled by the equation below:

T=68−1.25tT = 68 - 1.25t

Where 0≤t≤240 \leq t \leq 24. Which of the following is the best interpretation of the number 6868 in this context?

  1. The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned onAnswer
  2. B
    The rate, in degrees Fahrenheit per minute, at which the temperature inside the compartment decreases
  3. C
    The temperature inside the compartment, in degrees Fahrenheit, 1 minute after the cooling system is turned on, which is 67.25∘F67.25^\circ\text{F}
  4. D
    The number of minutes it takes for the temperature inside the compartment to reach 0∘F0^\circ\text{F}

Answer

The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on
The correct answer correctly identifies the meaning of the constant term 6868 as the y-intercept of the linear equation. In this context, the y-intercept represents the value of TT when the time tt is 00, which corresponds to the initial temperature inside the compartment before any cooling occurs.

Step-by-Step Solution

1
Identify the form of the linear equation and its components.
The equation is in the slope-intercept form, T=mt+bT = mt + b, where m=−1.25m = -1.25 is the slope and b=68b = 68 is the TT-intercept (or vertical intercept).
Recognizing the mathematical meaning of each coefficient is the first step in interpreting them in a real-world context.
2
Determine the physical meaning of the vertical intercept (b=68b = 68).
The vertical intercept occurs when the independent variable, tt (time in minutes), is equal to 00. Evaluating TT at t=0t = 0 gives T(0)=68−1.25(0)=68T(0) = 68 - 1.25(0) = 68.
Setting the independent variable to zero allows us to define the starting state of the system.
3
Interpret this starting state in the context of the problem.
At t=0t = 0, the cooling system has just been turned on. Thus, 68∘F68^\circ\text{F} represents the temperature inside the truck's cooling compartment when the system is first turned on.
Connecting the value of T(0)T(0) back to the scenario provides the correct interpretation of the constant 6868.

Key Concept

Interpreting the y-intercept of a linear model in context
Estimated Time:1m 30s
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Topics

Interpreting Linear Relationships in Context

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