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Difficulty: MediumInterpreting Linear Relationships in Context

A digital archiving team is scanning a large collection of historical documents. The number of unscanned documents remaining in the collection, DD, can be modeled by the equation D=14200−350tD = 14{}200 - 350t, where tt represents the number of hours the team has been scanning. Which of the following is the best interpretation of 350350 in this context?

  1. A
    The number of hours it takes the team to scan one document
  2. B
    The initial number of unscanned documents at the start of the shift
  3. The number of documents the team scans each hourAnswer
  4. D
    The number of unscanned documents remaining after one hour of the shift

Answer

The number of documents the team scans each hour
In the linear model D=14200−350tD = 14{}200 - 350t, the coefficient −350-350 is the slope of the line, which indicates the rate of change of the dependent variable. Since the remaining number of unscanned documents decreases by 350350 for each hour that passes, the value 350350 represents the number of documents scanned per hour by the team.

Step-by-Step Solution

1
Identify the role of each component in the linear equation D=14200−350tD = 14{}200 - 350t.
The equation is in the form y=b+mxy = b + mx, where the constant b=14200b = 14{}200 is the y-intercept (the starting value of DD when t=0t = 0) and the coefficient m=−350m = -350 is the slope (the constant rate of change of DD per unit increase in tt).
Understanding the structure of the linear model helps distinguish between initial values and rates of change.
2
Interpret the meaning of the slope in the context of the variables.
The slope of −350-350 indicates that for each unit increase in the independent variable tt (hours), the dependent variable DD (unscanned documents remaining) decreases by 350350.
The slope represents the constant rate of change of the remaining documents over time.
3
Select the option that matches this rate of change.
A rate of 350350 documents per hour means the archiving team scans 350350 documents every hour.
This directly matches the definition of the slope in this context.

Key Concept

Interpreting Linear Relationships in Context
Estimated Time:1m 15s
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Interpreting Linear Relationships in Context

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