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Difficulty: HardInterpreting Linear Relationships in Context

An IT administrator uses the equation T=0.06r+40T = 0.06r + 40 to model the total time TT, in milliseconds, required to retrieve rr records from a cloud database. The constant term in the model represents the total setup time, which consists of network latency and database connection overhead. If the database connection overhead is 20% of the total setup time, which of the following is the best interpretation of the value 8 in this model?

  1. A
    The increase in database connection overhead, in milliseconds, for each additional record retrieved.
  2. B
    The network latency, in milliseconds, for retrieving any number of records.
  3. The database connection overhead, in milliseconds, for retrieving any number of records.Answer
  4. D
    The rate at which the total time increases, in milliseconds per record, for the first 8 records retrieved.

Answer

The database connection overhead, in milliseconds, for retrieving any number of records.
In the linear model T=0.06r+40T = 0.06r + 40, the constant term of 4040 represents the total setup time in milliseconds, which is independent of the number of records rr retrieved. The database connection overhead is 20%20\% of this total setup time, which equals 0.20×40=80.20 \times 40 = 8 milliseconds. Since it is part of the constant term, it represents a fixed overhead of 88 milliseconds for retrieving any number of records.

Step-by-Step Solution

1
Identify the constant term in the linear model T=0.06r+40T = 0.06r + 40.
The constant term (y-intercept) is 4040, which represents the total setup time of 4040 milliseconds when r=0r = 0 records are retrieved.
In a linear relationship of the form y=mx+by = mx + b, the constant term bb represents the initial value or y-intercept.
2
Calculate the database connection overhead.
The database connection overhead is 20%20\% of the 4040 milliseconds total setup time, which is 0.20×40=80.20 \times 40 = 8 milliseconds.
The problem states that the database connection overhead is 20% of the total setup time.
3
Interpret the calculated value of 8 in the context of the model.
Since the 4040 milliseconds is a constant term that does not depend on the number of records rr, the 88 milliseconds represents a fixed database connection overhead for any number of records retrieved.
The constant term in a linear model represents a fixed value that does not scale with the independent variable.

Key Concept

Interpreting the constant term (y-intercept) and its components in a linear relationship context.
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Interpreting Linear Relationships in Context

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