Question

微积分题目！ calculus

A swimming pool is 40ft long, 10 ft wide, 4ft deep at the shallow end and 9 ft deep at deep end, the bottom is an inclined plane. assume that the water is being pumped into the pool at 10ft^3/min and that there is 4ft of water at the deep end at the begining. 1. what is the percentage of the pool is filled? 2. At what rate is the water level rising.

Answer 1

the volume of the pool is:
40x10x4+40x(9-4)x10/2 = 2600 ft^3，it is the sum of a rectangular solid and a wedge.
the voume of water at the begining is:
(4/5)x40x(9-4)x10/2 = 800 ft^3
the volume of the wedge is: 40x(9-4)x10/2 = 1000 ft^3
so the time it takes to fill the wedge is: (1000-800)/10 = 20 min
1. the percentage of the pool is filled

100% x (10t+800)/2600 = (100/26)t +400/13, t is time in minutes
2. the depth of water is H
when 0<t<20
volume of water is V = 10t+800 = (1/2)xHx40x(H/5)x10 = 40H^2
H = sqrt[(1/4)t+20] ==> dH/dt = {(1/2)x1/sqrt[(1/4)t+20]}xd[(1/4)t+20]/dt = 1/{8sqrt[(1/4)t+20]}
sqrt is square root
when t>20
volume of water is V = 10t+1000 = (H-5)x40x10+1000

H = t/40 + 5 ==> dH/dt = 1/40 ft/min

Answer 2

咋说。。。