Chapter 1.5, Problem 40PS

Solution

To compare: the wind speeds required to generate 5 foot waves and 20 foot waves.

The wind speeds required to generate 5 foot waves is 16.2 knots and for the 20 foot waves, it is 32.4 knots.

Given:

The equation $h=0.019s^2$ gives the height h (in feet) of the largest ocean waves when the wind speed is s knots.

Property Used:

  $\begin{array}{c}\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}\\ \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\end{array}$

Calculation:

The equation $h=0.019s^2$ gives the height h (in feet) of the largest ocean waves when the wind speed is s knots.

First, let the height of ocean wave is 5 foot, then substituting $h=5$ in the given equation,

  $5=0.019s^2$ .

Divide both sides by $0.019$ .

  $s^2=\frac{5}{0.019}$ .

Take square root of both sides,

  $s=\pm \sqrt{\frac{5}{0.019}}$ .

Since, s represents speed it is a non negative quantity. Thus, the solution is:

  $\begin{array}{c}s=\sqrt{\frac{5}{0.019}}\\ \approx 16.2\text{ knots}\end{array}$

Hence, to generate 5 foot waves, wind speed required is 16.2 knots.

Now, let the height of ocean wave is 20 foot, then substituting $h=20$ in the given equation,

  $20=0.019s^2$ .

Divide both sides by $0.019$ .

  $s^2=\frac{20}{0.019}$ .

Take square root of both sides,

  $s=\pm \sqrt{\frac{20}{0.019}}$

Since, s represents speed it is a non negative quantity. Thus, the solution is:

  $\begin{array}{c}s=\sqrt{\frac{20}{0.019}}\\ \approx 32.4\text{ knots}\end{array}$

Hence, to generate 20 foot waves, wind speed required is 32.4 knots.