Chapter 4, Problem 102RE

a.

To determine

To describe: The change in the graph of the given function (without graphing the resulting function) when $A$ is changed from $\frac{1}{5}\text{to}\frac{1}{3}$ .

Answer to Problem 102RE

The maximum displacement at which the weight can be lifted will be changed from $\frac{1}{5}\text{ft}\text{to}\frac{1}{3}\text{ft}$ .

Explanation of Solution

Given Information: Given equation representing the model is $y=A{e}^{-kt}\cos bt=\frac{1}{5}{e}^{-t/10}\cos 6t$ , where $y$ is the distance (in feet) from equilibrium and $t$ is the time (in seconds) and a figure representing the model is given.

When $A$ is changed from $\frac{1}{5}to\frac{1}{3}$ , then the maximum displacement at which the weight can be lifted will be changed from $\frac{1}{5}ftto\frac{1}{3}ft$ .

b.

To determine

To describe: The change in the graph of the given function (without graphing the resulting function) when $k$ is changed from $\frac{1}{10}to\frac{1}{3}$ .

Answer to Problem 102RE

The number of oscillations before equilibrium will decrease as $e^{-t/3}<e^{-t/10}$ .

Explanation of Solution

Given Information: Given equation representing the model is $y=A{e}^{-kt}\cos bt=\frac{1}{5}{e}^{-t/10}\cos 6t$ , where $y$ is the distance (in feet) from equilibrium and $t$ is the time (in seconds) and a figure representing the model is given.

When $k$ is changed from $\frac{1}{10}to\frac{1}{3}$ , then the number of oscillations before equilibrium will decrease as $e^{-t/3}<e^{-t/10}$ .

c.

To determine

To describe: The change in the graph of the given function (without graphing the resulting function) when $b$ is changed from $6to9$ .

Answer to Problem 102RE

The period of oscillation will decrease and the system will reach the equilibrium faster.

Explanation of Solution

Given Information: Given equation representing the model is $y=A{e}^{-kt}\cos bt=\frac{1}{5}{e}^{-t/10}\cos 6t$ , where $y$ is the distance (in feet) from equilibrium and $t$ is the time (in seconds) and a figure representing the model is given.

When $b$ is changed from $6to9$ , then the period of oscillation will decrease and the system will reach the equilibrium faster.