As part of a quality-control program, the chess sets...

As part of a quality-control program, the chess sets manufactured by Jones Brothers are subjected to a final inspection before packing. The rate of increase in the number of sets checked per hour by an inspector $t$ hr into the 8 A.M. to 12 noon shift is approximately $$N^{\prime}(t)=-3 t^{2}+12 t+45 \quad(0 \leq t \leq 4)$$ a. Find an expression $N(t)$ that approximates the number of sets inspected at the end of $t$ hours. b. How many sets does the average inspector check during a morning shift?

As part of a quality-control program, the chess sets manufactured by Jones Brothers are subjected to a final inspection before packing. The rate of increase in the number of sets checked per hour by an inspector $t$ hr into the 8 A.M. to 12 noon shift is approximately
$$N^{\prime}(t)=-3 t^{2}+12 t+45 \quad(0 \leq t \leq 4)$$
a. Find an expression $N(t)$ that approximates the number of sets inspected at the end of $t$ hours.
b. How many sets does the average inspector check during a morning shift?

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Key Concepts

Average Value of a Function

The average value of a function over an interval is found by dividing the integral of the function over that interval by the length of the interval. This method is key to determining the mean number of items inspected per hour, providing an overall measure of performance during the shift.

Antiderivative and Indefinite Integration

This concept involves reversing the differentiation process to find a function from its rate of change. When given a derivative, integration is used to obtain the original function up to an additive constant, which is crucial for reconstructing the accumulated quantity over time.

Initial Value Problems

An initial value problem provides a condition that specifies the value of the function at a particular point. This additional information allows for the determination of the constant of integration, ensuring that the computed function accurately reflects the situation at the beginning of the observation period.

Definite Integration

Definite integration is used to compute the total accumulation of a quantity over a specific interval. In the context of the problem, it helps in calculating the total number of items inspected during an entire shift by integrating the rate function over that time period.

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