Chapter 13.5, Problem 57PPS

To determine

To find: the first positive time when the point 0.5 meter from the left end has a displacement of 0.01 millimeter.

Answer to Problem 57PPS

  $t\approx 0.0026\text{seconds}$ .

Explanation of Solution

Given information: A wave traveling in a guitar string can be modeled by the equation

  $D=0.5\sin (6.5x)\sin (2500t)$ , where D is the displacement in millimeter at the position x millimeters from the left end of the string at time t seconds.

Calculation:

Have to find t , when given D, x:

  $\begin{array}{c}D=0.5\sin (6.5x)\sin (2500t)\\ D=0.01\\ x=0.5\text{meter}=0.5\times 1000\text{millimeters}\\ 0.01=0.5\sin (6.5\times 0.5\times 1000)\sin (2500t)\\ 1=50\sin (3250)\sin (2500t)\\ \sin (2500t)=\frac{1}{50\sin (3250)}\\ 2500t={\sin }^{-1}(\frac{1}{50\sin (3250)})\\ t\approx 0.0026\text{seconds}\end{array}$

Therefore, $t\approx 0.0026\text{seconds}$ .