1. ## Related Rates Problem

Hi,I'm having some trouble with a related rate problem. A pool is 40ft long, 20 ft wide, 4 ft deep at the shallow end and 9 ft deep at the deep end. Water is being pumped into the pool at 10 ft^3/min. How fast is the water level rising when there is 4 ft of water in the deep end?

I've tried using volume of a trapezoidal prism and differentiating that, but it didn't seem to work. Any help is appreciated. Thanks a lot.

2. Since the question is for dh/dt when h=4ft at the 9-ft deep end, then the volume of a trapezoidal prism won't work. Because at h=4ft, you only have a triangular prism for the volume of water in the tank.

So,
Draw the figure, a longitudinal section of the pool, such that the walls are vertical. It is a trapezoid, (trapezium).
Draw horizontal line passing through the 4-ft deep.
Draw another horizontal line lower than tha first line.
Two similar right triangles are formed.

One has:
----vertical leg = (9-4) = 5ft
----horizontal leg = 40ft
----hypotenuse---unknown.

The smaller one has:
---vertical leg = h
---horizontal leg = x
---hypotenuse----unknown too.

By proportion,
5/40 = h/x
Cross multiply,
5x = 40h
x = 40h/5 = 8h

So, volume of water in the pool when depth at deep end is h ft,
V = (1/2)(h)(x) *(20)
V = 10h(8h)
V = 80h^2

dV/dt = 80(2h *dh/dt)
dV/dt = 160h (dh/dt)

When h=4, and dV/dt is always 10 cuft/min,
10 = 160(4)(dh/dt)
1 = 64(dh/dt)
dh/dt = 1/64

Therefore, at that instant, the water is rising at the rate of 1/64 ft/min. ----answer.

3. Hello, HclBr!

ticbol did an excellent job of explaining the solution.

Here are some diagrams . . .

This is the entire pool.
Code:
                         40
*-----------------------------*
20 /                             / |
/                             /   |4
*-----------------------------*     |
|                             |     *
4|                            4|   /
|                             | /
+ - - - - - - - - - - - - - - *
|                       *
5|                 *
|           *
|     *
*

This is the lower portion with the water.
Code:
                          40
*-----------------------------------------------*
|                                         *
|            x                      *
| - - - - - - - - - - - - - - *
5 | : : : : : : : : : : : *
|h : : : : : : : :*
| : : : : : *
|: : :*
*

### Dv/dt trapezoid prism pool

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