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Difficulty: HardIntegers, Absolute Value, and Number Lines

A scientist monitors the temperature in three different chambers, AA, BB, and CC. The temperature in Chamber BB is T∘CT^\circ\text{C}. The temperature in Chamber AA is always 5∘C5^\circ\text{C} colder than the temperature in Chamber BB, and the temperature in Chamber CC is always 9∘C9^\circ\text{C} warmer than the temperature in Chamber BB. If TT must be an integer, and the sum of the absolute values of the temperatures in all three chambers is minimized, what is the value of TT?

  1. A
    -9
  2. B
    -1
  3. 0Answer
  4. D
    2
  5. E
    5

Answer

The correct answer is 00.
The correct answer is the value 00. The sum of the absolute values of the temperatures in all three chambers is represented by the function S(T)=∣T−5∣+∣T∣+∣T+9∣S(T) = |T - 5| + |T| + |T + 9|. Geometrically, this sum represents the total distance from a point TT to the coordinates 55, 00, and −9-9 on a number line. For any set of points, the sum of absolute differences is minimized at their median. The median of the set {−9,0,5}\{-9, 0, 5\} is 00. Evaluating the sum at T=0T = 0 yields ∣0−5∣+∣0∣+∣0+9∣=5+0+9=14|0 - 5| + |0| + |0 + 9| = 5 + 0 + 9 = 14. Any other integer choice yields a larger sum (for example, a value of 11 or −1-1 yields a sum of 1515).

Step-by-Step Solution

1
Write expressions for the temperatures of the three chambers in terms of TT.
The temperature in Chamber AA is T−5T - 5, in Chamber BB is TT, and in Chamber CC is T+9T + 9.
To represent the temperature of each chamber relative to Chamber BB.
2
Write the sum of the absolute values of the three temperatures.
The sum is represented by the expression S(T)=∣T−5∣+∣T∣+∣T+9∣S(T) = |T - 5| + |T| + |T + 9|.
To formulate the function that needs to be minimized.
3
Find the key values that make each absolute value term zero.
The points are 55, 00, and −9-9. Ordered on a number line, these points are −9-9, 00, and 55.
The sum of absolute differences is geometrically equivalent to the sum of distances on a number line, which is minimized at the median of the points.
4
Evaluate the sum of absolute values at the median value, T=0T = 0, and compare it to other values.
For T=0T = 0, S(0)=∣−5∣+∣0∣+∣9∣=14S(0) = |-5| + |0| + |9| = 14. For T=1T = 1, S(1)=∣−4∣+∣1∣+∣10∣=15S(1) = |-4| + |1| + |10| = 15. For T=−1T = -1, S(−1)=∣−6∣+∣−1∣+∣8∣=15S(-1) = |-6| + |-1| + |8| = 15. Thus, the minimum sum occurs at T=0T = 0.
To verify that the median yields the minimum sum of absolute values.

Key Concept

Minimizing the sum of absolute values of linear terms by finding the median of their zero-points on a number line.
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Topics

Integers, Absolute Value, and Number Lines

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