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 $8 \mathrm{a} . \mathrm{m}$. to 12 noon morning 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. Hint: $N(0)=0$. 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 $8 \mathrm{a} . \mathrm{m}$. to 12 noon morning 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. Hint: $N(0)=0$.
b. How many sets does the average inspector check during a morning shift?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Average Value of a Function

The average value of a function over an interval is calculated by dividing the definite integral of the function over that interval by the interval's length. This concept is used to find a representative value that summarizes the function's behavior throughout the interval.

Definite Integration

Definite integration involves computing the integral of a function over a specified interval. It is used to determine the net accumulation or total change of a quantity over that interval, based on its rate of change.

Initial Condition

In the context of solving differential equations, an initial condition is a known value of the function at a specific point. It allows us to determine the constant of integration when computing an antiderivative, thereby obtaining a unique solution that fits a given scenario.

Antiderivative

An antiderivative is a function whose derivative equals the given function. This concept is central when recovering an original function from a rate of change expression by performing the inverse operation of differentiation, known as integration.

Transcript

Please give Ace some feedback