The intensity $\mathrm { L } ( \mathrm { x } )$ of light $\mathrm { x } \mathrm { ft }$ beneath the surface of a lake satisfies the differential equation $\frac { \mathrm { dL } } { \mathrm { dx } } = - 0.05 \mathrm {~L}$. At what depth, to the nearest foot, is the intensity one tenth the intensity at the surface?
A) 46 ft
B) 31 ft
C) 23 ft
D) 69 ft

Question

Quizplus · Accepted Answer

The intensity is one tenth the intensity at the surface when $L(x) = \frac{1}{10} L(0)$. Solving the differential equation gives $L(x) = L(0) e^{-0.05x}$. So, we have $e^{-0.05x} = \frac{1}{10}$, which gives $x = \frac{\ln 10}{0.05} \approx 46$.

The Intensity L(x)\mathrm { L } ( \mathrm { x } )L(x)

The Intensity $\mathrm { L } ( \mathrm { x } )$