Chapter 7, Problem 57RE

To determine

To Calculate:

To calculate how deep the diver can work without artificial light.

Answer to Problem 57RE

The diver can work for about $60$ feet deep without artificial light.

Explanation of Solution

Given information:

The intensity $L\left(x\right)$ of light $x$ feet beneath the surface of the ocean satisfies the differential equation

  $\frac{dL}{dx}=-kL$ where $k$ -constant

Diver can diving to $18$ feet in Caribbean sea cuts the intensity in half.

Formula:

The intensity $L\left(x\right)$ of light $x$ feet beneath the surface of the ocean satisfies the differential equation

  $\frac{dL}{dx}=-kL$ where $k$ -constant

Modelling growth with other bases $y=y_0{e}^{kx}$

Where $y_0$ - value of $y$ at $t=0$ ; $k$ -half life of the substance.

The intensity $L\left(x\right)$ of light $x$ feet beneath the surface of the ocean satisfies the differential equation

  $\frac{dL}{dx}=-kL$ where $k$ -constant

  $\begin{array}{l}\frac{dL}{dx}=-kL\\ \frac{dL}{L}=-kdx\end{array}$

Differentiating with respect to x,

  $\begin{array}{l}\ln \left|L\right|=-kx+C\\ L=L_0{e}^{-kx}\\ 0.5=e^{-k\cdot 18}\\ k=0.0385\end{array}$

So let $y=0.1y_0$ , $k=0.0385$ ,

Now substituting this in modelling growth with other bases,

  $\begin{array}{l}y=y_0{e}^{kx}\\ 0.1y_0=y_0{e}^{0.0385x}\\ 0.1=e^{0.0385x}\\ x=59.8\end{array}$

Therefore, the diver can work for about $60$ feet deep without artificial light.