Chapter 1.2, Problem 17E

(a)

To determine

To express: The water pressure in terms of the depth below the ocean surface.

Answer to Problem 17E

The equation of the water pressure in terms of the depth below the ocean surface is $p=0.434d+15$.

Explanation of Solution

Let p represents the pressure d represents the depth.

Since the water pressure below the surface of the ocean is increases by $4.34{\text{ lb/in}}^{\text{2}}$ for every 10 feet of depth, the slope (m) is calculated as follows.

$\begin{array}{c}m=\frac{\text{Change in pressure}}{\text{change in depth}}\\ =\frac{4.34}{10}\\ =0.434\end{array}$

Therefore, the slope is, $m=0.434$.

Also, it is given that the water pressure at the surface of the ocean is same as the air pressure above the water, $15{\text{ lb/in}}^{\text{2}}$.

This can be expressed as the point (0, 15) where 0 represents the depth and the 15 represents the water pressure.

Recall the point-slope form $y-y_1=m\left(x-x_1\right)$ where m is the slope and $\left(x_1,y_1\right)$ is the point.

Substitute the slope $m=0.434$ and the point $\left(x_1,y_1\right)=\left(0,15\right)$ in slope-point form and obtain the equation of pressure as a function of depth.

$\begin{array}{l}p-15=0.434\left(d-0\right)\\ p=0.434d+15\end{array}$

Thus, the required equation is $p=0.434d+15$.

(b)

To determine

To find: The depth when the pressure is $100{\text{ lb/in}}^{\text{2}}$.

Answer to Problem 17E

The depth is approximately 196 feet when the water pressure is $100{\text{ lb/in}}^{\text{2}}$.

Explanation of Solution

From part (a), the equation of the water pressure as the function of the depth is $p=0.434d+15$.

Substitute $p=100$ and obtain the value of d as follows.

$\begin{array}{l}100=0.434d+15\\ 100-15=0.434d\\ 85=0.434d\\ d\approx 196\end{array}$

Thus, the water pressure is $100{\text{ lb/in}}^{\text{2}}$ at the depth about 196 feet.