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Difficulty: MediumFunction Evaluation, Composition, and Properties

A temperature control system in a chemical reactor regulates target output temperature, in degrees Celsius, based on the reactor's pressure coefficient. The pressure coefficient PP is determined by the elapsed time tt, in minutes, according to the function P(t)=(t−3)2−5P(t) = (t - 3)^2 - 5. The target temperature TT is determined by the pressure coefficient PP according to the function T(P)=3∣P−2∣+1T(P) = 3|P - 2| + 1. What is the target output temperature of the system, in degrees Celsius, at time t=1t = 1 minute?

  1. A
    -8
  2. B
    12
  3. 10Answer
  4. D
    -4
  5. E
    46

Answer

10
The correct temperature is 10. By first finding the pressure at t=1t = 1 to be P(1)=(1−3)2−5=4−5=−1P(1) = (1 - 3)^2 - 5 = 4 - 5 = -1, and then substituting this pressure into the temperature function, we get T(−1)=3∣−1−2∣+1=3(3)+1=10T(-1) = 3|-1 - 2| + 1 = 3(3) + 1 = 10.

Step-by-Step Solution

1
Calculate the pressure coefficient P(1)P(1) at time t=1t = 1 minute by substituting t=1t = 1 into P(t)=(t−3)2−5P(t) = (t - 3)^2 - 5.
P(1)=−1P(1) = -1
Before calculating the temperature, we must find the input value PP corresponding to t=1t = 1.
2
Evaluate the target temperature T(P)T(P) at the pressure coefficient P=−1P = -1 by substituting P=−1P = -1 into T(P)=3∣P−2∣+1T(P) = 3|P - 2| + 1.
T(−1)=10T(-1) = 10
Applying the absolute value definition ∣−3∣=3|-3| = 3, we find 3(3)+1=103(3) + 1 = 10.

Key Concept

Evaluating composite functions in a real-world scenario.

Alternative Method

Instead of evaluating step-by-step numerically, we can write the composite function symbolically: T(P(t))=3∣((t−3)2−5)−2∣+1=3∣(t−3)2−7∣+1T(P(t)) = 3|((t - 3)^2 - 5) - 2| + 1 = 3|(t - 3)^2 - 7| + 1. Plugging in t=1t = 1 yields T(P(1))=3∣(1−3)2−7∣+1=3∣4−7∣+1=3∣−3∣+1=10T(P(1)) = 3|(1 - 3)^2 - 7| + 1 = 3|4 - 7| + 1 = 3|-3| + 1 = 10.
Estimated Time:1m 30s
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Topics

Function Evaluation, Composition, and Properties

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