exponential decay/growth?

Question

the fact thatthe intesity of light falls exponentially with depth in water. Suppose that at a depth of 25 ft the water absorbs 15% of the light that strikes the surface. At what depth would the light at noon be as bright as a full moon, which is one three-hundred-thousandth as bright as the noonday sun?

Depth in feet=?

I have been working at this all night and I cant seem to figure it out. If somebody could explain it. Incase you forgot, exponential growth is A(t)=Ao e^(kt). or y'=ky

Answer 1

So here, t stands for depth. Then 85 = 100 e^(20k)

100 represents 100% and 85 represents what's left after 15% of that 100% gone

0.85 = e^(20k)

ln (0.85) = ln e^(20K) = 20K

k = ln(0.85)/20 = -0.008.....

(1/3000) (100) = 100 e^(-0.008...t)

(1/3000) = e^(-0.008....t)

ln (1/3000) = (-0.008...t)

ln(1/3000) / -0.008... = t

I got 985 feet for t

What do you think, does this make sense?

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Answer 2

♠ yes, a(t)=a0*exp(-kt); and first we have to find k;
and ln(a(t)/a0)=-kt, hence k=-ln(a(t)/a0) /t;
♣ 15%=0.15=(a0-a(t))/a0, hence a0*0.15=a0-a(t), a(t)=0.85*a0;
♦ so k=-ln(a(t)/a0) /t =-ln(0.85)/25 = 0.006501/ft;
♥ thus for any depth t a(t)/a0=exp(-0.006501*t);
? moon? 0.300=exp(-0.006501*t), ln(0.3)=-0.006501*t, t=185.2ft;