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Difficulty: MediumInterpreting Linear Relationships in Context

A team of glaciologists monitors the thickness of an alpine glacier during the summer season. The thickness of the glacier, TT, in meters, can be modeled by the equation T=54.5−0.12dT = 54.5 - 0.12d, where dd represents the number of days since the start of the summer season, for 0≤d≤1500 \leq d \leq 150. Which of the following is the best interpretation of the number 0.120.12 in this context?

  1. A
    The initial thickness of the glacier, in meters, at the start of the summer season
  2. B
    The number of days it takes for the thickness of the glacier to decrease by 11 meter
  3. The decrease in the thickness of the glacier, in meters, each day during the summer seasonAnswer
  4. D
    The total number of days from the start of the summer season until the glacier melts completely

Answer

The correct answer is the option stating that 0.12 is the decrease in the thickness of the glacier, in meters, each day during the summer season.
The linear model is given by T=54.5−0.12dT = 54.5 - 0.12d, which is in the slope-intercept form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. In this equation, the slope mm is −0.12-0.12, which represents the rate of change of the glacier's thickness per day. Since the slope is negative, the thickness decreases by 0.120.12 meters each day. Therefore, the number 0.120.12 represents the decrease in the thickness of the glacier, in meters, each day during the summer season.

Step-by-Step Solution

1
Identify the components of the linear equation T=54.5−0.12dT = 54.5 - 0.12d.
The constant term is 54.554.5, and the coefficient of the variable dd is −0.12-0.12.
To interpret a linear relationship in context, we must distinguish between the initial value (y-intercept) and the rate of change (slope).
2
Determine the meaning of the slope in this context.
The slope is −0.12-0.12 meters per day, meaning the glacier's thickness TT decreases by 0.120.12 meters for each day dd that passes.
The coefficient of the independent variable in a linear equation represents the rate of change of the dependent variable per unit of the independent variable.
3
Relate the number 0.120.12 to the rate of change.
The positive value 0.120.12 represents the magnitude of this rate of change, which is the amount of decrease in thickness per day.
Since the slope is negative, the change is a decrease, and the rate of this decrease is 0.120.12 meters per day.

Key Concept

Interpreting the slope of a linear relationship in a real-world context
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Topics

Interpreting Linear Relationships in Context

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