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Difficulty: EasyInterpreting Linear Relationships in Context

At a local coffee shop, the remaining weight of coffee beans CC, in pounds, after preparing nn cups of espresso is modeled by the equation C=80−0.04nC = 80 - 0.04n. What is the best interpretation of the number 0.040.04 in this context?

  1. The weight of coffee beans, in pounds, used to prepare each cup of espressoAnswer
  2. B
    The initial weight of the coffee beans, in pounds, before any cups of espresso are prepared
  3. C
    The number of cups of espresso that can be prepared per pound of coffee beans
  4. D
    The weight of coffee beans, in pounds, remaining in the shop after preparing 8080 cups of espresso

Answer

The weight of coffee beans, in pounds, used to prepare each cup of espresso
The correct answer is the option stating that 0.040.04 is the weight of coffee beans, in pounds, used to prepare each cup of espresso. In the linear relationship C=80−0.04nC = 80 - 0.04n, the slope is −0.04-0.04, which represents the rate of change of the remaining weight of coffee beans with respect to the number of cups of espresso prepared. A rate of −0.04-0.04 pounds per cup means that for every cup prepared, the remaining coffee beans decrease by 0.040.04 pounds, indicating that 0.040.04 pounds of beans are consumed per cup.

Step-by-Step Solution

1
Analyze the structure of the linear equation C=80−0.04nC = 80 - 0.04n.
The equation is in the slope-intercept form y=mx+by = mx + b, where the dependent variable is CC (remaining coffee beans in pounds), the independent variable is nn (number of cups of espresso), the constant (y-intercept) is 8080, and the slope (coefficient of nn) is −0.04-0.04.
Identifying the components of the linear equation helps in assigning their real-world contextual meanings.
2
Interpret the meaning of the slope in the context of the variables.
The slope of −0.04-0.04 represents the change in the remaining coffee beans (CC) for every 11-unit increase in the number of cups of espresso (nn). This means the remaining weight decreases by 0.040.04 pounds per cup.
The slope of a linear model shows the constant rate of change of the dependent variable per unit of the independent variable.
3
Relate the rate of decrease to the options.
A decrease of 0.040.04 pounds of remaining beans per cup of espresso means that 0.040.04 pounds of coffee beans are used to prepare each cup of espresso.
Translating the decrease rate into consumption rate matches the physical scenario described.

Key Concept

Interpreting Linear Relationships in Context
Estimated Time:45s
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Topics

Interpreting Linear Relationships in Context

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