Water leaks out of a barrel at a rate proportional to the square root of the dep

Question

Water leaks out of a barrel at a rate proportional to the square root of the depth, 0, of the water at that time. Depth is measured in inches and time is measured in hours. Write a differential equation for the depth of water at time t. Assume the proportionality constant is k > 0. dD / dt If the depth of the water is initially 36 inches, find D(t). Your answer will have k in it. 0(t) = If the depth of the water drops to 35 inches in 1 hour, how long will it take for all the water to leak out of the barrel? Exact answer

Explanation / Answer

1)dD/dt = ksqrt(D)

2) therefore dD/sqrt(D) = kdt

integrating

2 D0.5 = kt+c (c is integration constant

given at t=0 D is 36

substituing we get c = 12

D(t) = (kt+12)2/4

3) 35 = (k+12)2/4

k = -0.16784

D^0.5 = kt+12/2

D=0

0 = -0.16784t +12

t = 12/0.16784 =71.496

= 71.5 hours

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