The rate at which a machine operator's efficiency, E...

The rate at which a machine operator's efficiency, $E$ (expressed as a percentage), changes with respect to time $t$ is given by $\frac{d E}{d t}=40-10 t$ where $t$ is the number of hours the operator has been at work. a) Find $E(t),$ given that the operator's efficiency after working 2 hr is $72 \% ;$ that is, $E(2)=72$ b) Use the answer to part (a) to find the operator's efficiency after $4 \mathrm{hr}$; after $8 \mathrm{hr}$.

The rate at which a machine operator's efficiency, $E$ (expressed as a percentage), changes with respect to time $t$ is given by $\frac{d E}{d t}=40-10 t$
where $t$ is the number of hours the operator has been at work.
a) Find $E(t),$ given that the operator's efficiency after working 2 hr is $72 \% ;$ that is, $E(2)=72$
b) Use the answer to part (a) to find the operator's efficiency after $4 \mathrm{hr}$; after $8 \mathrm{hr}$.

Key Concepts

First-Order Differential Equations

A first-order differential equation involves the first derivative of a function and is used to model rates of change. In a typical problem, the equation relates the derivative of an unknown function to the independent variable, and solving it involves finding a function that satisfies this rate of change over time or another variable.

Integration as an Antiderivative

Integration is the process of finding the antiderivative of a function, which essentially reverses differentiation. When given the rate of change of a function, integrating that rate yields the original function, up to an arbitrary constant, which represents the family of all possible antiderivatives.

Initial Value Problems

An initial value problem (IVP) is a differential equation accompanied by given initial conditions. The initial condition specifies the value of the unknown function at a particular point, which allows for the determination of a unique solution from the general solution by solving for the constant of integration.

Use of Initial Conditions to Determine Constants

When a general solution to a differential equation includes an arbitrary constant from the integration process, an initial condition is used to compute this constant. Substituting the specific value of the function and its corresponding independent variable into the general solution allows for the precise determination of this constant, which in turn yields the particular solution to the problem.

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