Filling Swimming Pool Related Rates Problem

Question

Spring in Southern California has finally arrived. You are looking forward to filling your swimming pool, which has a length of 10 meters and ends with trapezoidal shape. The trapezoids on the ends have a bottom edge with length 6 meters, a top edge with length 12 meters, and a height of 2 meters. Your hose can produce 3 cubic meters of water per hour. You begin to fill your pool. When the height of the water reaches 0.1 m, you realize this will take a long time. Calculate the rate the height of the water is increasing when it is 0.1 m high. Use this value to estimate how long it will take to fill the pool. Is this an overestimate or underestimate? Explain your reasoning.

Can you please give a step by step explanation that would be much appreciated, thank you!

Answer 1

Volume=Area of trapezoid*length=1/2(b1+b2)h*L

Volume of Water=1/2(6+w)h*10

Similar trapezoids:h/w=2/12

12h=2w

6h=w

Replace w in our volume formula with 6h

V=1/2(6+6h)h*12=1/2*(6h+6h²)*10 =30h+30h²

^{Take derivative of formula with respect to time t}

^{dV/dt = 30dh/dt + 60h*dh/dt}

^{Plug in for dV/dt and h, solve for dh/dt}

^{3 = 30dh/dt+60(0.1)dh/dt}

^{3 = 30dh/dt+6dh/dt}

^{3 = 36 dh/dt}

^{1/12 = dh/dt}

^{The height of the water in the pool is rising at the rate of 1/12 m per hr.}

^{Amount = Rate x Time}

Height = dh/dt x Time

2=(1/12)T

24 hrs=T

Underestimate because the pool is getting wider so the rate of change of the height (dh/dt) will be getting smaller.