Chapter 4.3, Problem 57E

Solution

a.

To identify: The displacement at the given time. Use Radian mode. $t=0$

A weight suspended from a spring is set into motion. Its displacement d from equilibrium is modeled by the equation where d is the displacement in inches and t is the time in seconds.

  $d=0.4\mathrm{c}os4t$

  

The displacement of the weight at $t=0$ is $\text{0}\text{.4}\text{inches}$

Given information:

The given equation is $d=0.4\mathrm{c}os4t$

  

Explanation:

Consider the given information.

Here the variable t is time in second and the displacement d is in inches.

The displacement of the weight for $t=0$

Substitute $t=0$ on the given equation

  $\begin{array}{c}d=0.4e^{-0.2t}\cos 4t\\ =0.4e^{-0.2\cdot 0}\cos \left(4\cdot 0\right)\\ =0.4\cdot e^0\cdot \cos 0\\ =0.4\cdot 1\cdot 1\\ =0.4\end{array}$

Therefore, the displacement of the weight at $t=0$ is $\text{0}\text{.4}\text{inches}$ .

b.

To identify: The displacement at the given time. Use Radian mode. $t=3$

The displacement of the weight at $t=3$ is $\text{0}\text{.1852}\text{inches}$

Given information:

The given equation is $d=0.4\mathrm{c}os4t$ .

  

Calculation:

Consider the given information.

Here the variable t is time in second and the displacement d is in inches.

The displacement of the weight for $t=3$

Substitute $t=3$ on the given equation

  $\begin{array}{c}d=0.4e^{-0.2t}\cos 4t\\ =0.4e^{-0.2\cdot 3}\cos \left(4\cdot 3\right)\\ =0.4\cdot e^{0.6}\cdot \cos 12\\ =0.4\cdot \left(0.548811\right)\left(0.84385\right)\\ \approx 0.1852\end{array}$

Therefore, the displacement of the weight at $t=3$ is $\text{0}\text{.1852}\text{inches}$ .