Francisca Rodden

2021-12-15

The intensity L(x) of light x feet beneath the surface of the ocean satisfies the differential equation dL/dx = - kL, where k is a constant. As a diver you know from experience that diving to 18 ft in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below a tenth of the surface value. About how deep can you expect to work without artificial light?

Durst37

Beginner2021-12-16Added 37 answers

$\frac{1}{L}dL=-kdx$

$\int \frac{1}{L}dL=-\int kdx$

Power Fotmula: $\frac{1}{u}du=\mathrm{ln}\left|u\right|+C$

$\mathrm{ln}\left|L\right|=-kx+C$

$e}^{\mathrm{ln}\left|L\right|}={e}^{-kx\cdot C$

$L=C{e}^{-kx}$

$0,5={e}^{-k\cdot 18}$

$\mathrm{ln}0.5={\mathrm{ln}e}^{-18k}$

$\mathrm{ln}0.5=-18k$

$k=-\frac{\mathrm{ln}0.5}{18}$

$k\approx 0.0385$

$0.1={e}^{-0.0385\cdot x}$

$\mathrm{ln}0.1={\mathrm{ln}e}^{-0.0385x}$

$\mathrm{ln}0.1=-0.0385$

$x=\frac{\mathrm{ln}0.1}{0.0385}$

$k\approx 59.8\text{}ft$

22+64

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