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Difficulty: Very hardInterpreting Linear Relationships in Context

A logistics company operates two distribution centers, Center A and Center B. The daily operating cost, CAC_A, in dollars, at Center A when processing pp packages is given by the equation CA=1.75p+3,200C_A = 1.75p + 3,200. The daily operating cost, CBC_B, in dollars, at Center B when processing pp packages is given by the equation CB=2.25(p−400)+2,800C_B = 2.25(p - 400) + 2,800, where p≥400p \geq 400. The difference in daily operating costs, DD, in dollars, between Center B and Center A is defined as D=CB−CAD = C_B - C_A. For p≥400p \geq 400, this difference is modeled by the equation D=0.50p−1,300D = 0.50p - 1,300. Which of the following is the best interpretation of the number 0.500.50 in this context?

  1. A
    For each additional package processed, the daily operating cost at Center A increases by 0.500.50 dollars more than the daily operating cost at Center B.
  2. For each additional package processed, the daily operating cost at Center B increases by 0.500.50 dollars more than the daily operating cost at Center A.Answer
  3. C
    For each increase of 2.002.00 dollars in the difference of daily operating costs between the two centers, Center B processes 11 additional package.
  4. D
    The cost to process each package at Center B is 0.500.50 dollars.

Answer

For each additional package processed, the daily operating cost at Center B increases by 0.500.50 dollars more than the daily operating cost at Center A.
The equation D=0.50p−1,300D = 0.50p - 1,300 models the difference in daily operating costs, DD, between Center B and Center A (CB−CAC_B - C_A) as a function of the number of packages processed, pp. The coefficient of pp, which is 0.500.50, represents the slope of this linear relationship. This means that for each unit increase in pp (each additional package processed), the difference DD increases by 0.500.50 dollars. Since D=CB−CAD = C_B - C_A, an increase in DD indicates that Center B's daily operating cost is increasing by 0.500.50 dollars more than Center A's daily operating cost for each additional package processed.

Step-by-Step Solution

1
Understand the meaning of the variables and the difference equation D=CB−CAD = C_B - C_A.
The variable pp represents the number of packages processed, and DD represents the difference in daily operating costs between Center B and Center A.
Establishing the relationship between the independent variable and the dependent variable is necessary to interpret the slope.
2
Identify the slope in the linear equation D=0.50p−1,300D = 0.50p - 1,300.
The equation is in slope-intercept form, y=mx+by = mx + b, where the slope mm is 0.500.50.
The slope represents the rate of change of the dependent variable DD with respect to the independent variable pp.
3
Interpret the rate of change in the context of the problem.
A slope of 0.500.50 means that for every increase of 11 in pp (each additional package processed), the value of DD increases by 0.500.50 dollars. Since D=CB−CAD = C_B - C_A, an increase of 0.500.50 in DD means that CBC_B (Center B's cost) increases by 0.500.50 dollars more than CAC_A (Center A's cost).
This links the mathematical rate of change directly to the real-world difference in costs between the two centers.

Key Concept

Interpreting the slope of a combined linear relationship in context
Estimated Time:1m 30s
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Topics

Interpreting Linear Relationships in Context

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