A swimming pool is 20𝑓𝑡20ft wide, 40𝑓𝑡40ft long,

Question

A swimming pool is 20𝑓𝑡20ft wide, 40𝑓𝑡40ft long, 3𝑓𝑡3ft deep at the shallow end, and 9𝑓𝑡9ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of .8𝑓𝑡3/𝑚𝑖𝑛.8ft3/min, how fast is the water level rising when the depth at the deepest point is 5𝑓𝑡5ft?

Answer 1

Let d be the depth of the water at the deep end.

The volume of water in the pool is a triangular wedge with a volume of

V = width•length•depth/2

until the depth d is 6 ft deep. After that, the volume increases by 40•20•(d-6). Since we only care about the depth up to 5 ft, we can use V = 20•L•d/2, where L is the length of the triangle as it it being filled. The relationship between the length and the depth, by similar triangles is L/d = 40/6 or L = 40d/6.

V = 20•(40/6)d²/2 = (200/3)d²

V' = (400/3)dd'

d' = (3/400)V'/d

when d = 5,

d' = (3/400)(0.8)/5 = 0.0012 ft/min