The intensity L(x) of light x feet beneath the surface of the ocean decreases at

Question

The intensity L(x) of light x feet beneath the surface of the ocean decreases at a rate proportional to its value at that location. That is, L(x) satisfies the differential equation dL/dx = -kL, for some k > 0 (the constant of proportionality). An experienced diver has determined that the weather conditions on the day of her dive will be such that the light intensity will be cut in half upon diving 20 ft under the surface of the water. She also knows that, once the intensity of the light falls below 1/6 of the surface value, she will have to make use of artificial light. How deep can the diver go without having to resort to the use of artificial light? ANSWER = ft. You have attempted this problem 0 times. You have 20 attempts remaining.

Explanation / Answer

given dL/dx=-kL

dL/L=-k dx

integrate on both sides

dL/L=-k dx

lnL=-kx+C

L=e-kx+C

L=Ce-kx

intensity is to the full on the surface, x =0

Lo=Ce-0

C=Lo

L=Loe-kx

light intensity will be cut in half upon diving 20 ft under the surface of the water

(1/2)Lo=Loe-k*20

e-k*20=(1/2)

-20k= ln(1/2)

20k =ln(2)

k=(1/20)ln2

k=0.034657359

L=Loe-0.034657359x

intensity of the light falls below 1/6 of the surface value, she will have to make use of artificial light

(1/6)Lo=Loe-0.034657359x

(1/6)=e-0.034657359x

-0.034657359x=ln(1/6)

0.035657359x=ln6

x=(ln6)/0.034657359

x=51.69925

x =51.7 ft

depth the diver can go without having to resort to the use of artificial light is 51.7 ft

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