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Difficulty: EasyInterpreting Linear Relationships in Context

A botanist tracks the growth of a bamboo plant. The height hh, in inches, of the plant dd days after the tracking began can be modeled by the equation h=4.5d+12h = 4.5d + 12. Which of the following is the best interpretation of 1212 in this context?

  1. A
    The height of the bamboo plant, in inches, 1 day after the tracking began.
  2. The initial height of the bamboo plant, in inches, when the tracking began.Answer
  3. C
    The average growth of the bamboo plant, in inches per day.
  4. D
    The number of days the tracking lasted.

Answer

The initial height of the bamboo plant, in inches, when the tracking began.
In the linear equation h=4.5d+12h = 4.5d + 12, the constant term 1212 represents the y-intercept. This corresponds to the value of the function hh when the independent variable dd is equal to 00. Since dd is the number of days since tracking began, d=0d = 0 is the start of tracking, meaning 1212 represents the initial height of the bamboo plant, in inches.

Step-by-Step Solution

1
Identify the component of the linear equation h=4.5d+12h = 4.5d + 12 to be interpreted.
The number to interpret is the constant term 1212.
We must determine the contextual meaning of the constant value in the linear model.
2
Find the mathematical meaning of the constant in the slope-intercept form y=mx+by = mx + b.
The constant term 1212 represents the y-intercept of the linear equation, which occurs when the independent variable d=0d = 0.
In a linear function, the constant represents the value of the dependent variable when the independent variable is zero.
3
Translate the mathematical definition to the real-world context of the problem.
Since dd represents the number of days after tracking began, d=0d = 0 represents the start of tracking. At d=0d = 0, h=12h = 12, which is the initial height of the bamboo plant in inches.
Relating the y-intercept to the context yields the initial state of the measured quantity.

Key Concept

Interpreting the y-intercept of a linear model in context
Estimated Time:45s
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Topics

Interpreting Linear Relationships in Context

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