Hello, Ron!

A swimming pool is 25 ft wide, 60 ft long, 3 ft deep at the shallow end, and 15 ft at its deepest point.

If the pool is being filled at a rate of 800 ft³/min, at what rate is the water level rising

when the depth at the deepest point is 5ft?

Side view of the pool . . . Ignore the top 3 feet. Code:

: - - - - 60 - - - - - :
- *-----------------------*
: | x *
: - +---------------*
12 : |:::::::::::*
: y |:::::::*
: : |:::*
- - *

The area of the water is: .$\displaystyle \,A\,=\,\frac{1}{2}xy$

From similar right triangles, we have: .$\displaystyle \,\frac{x}{y}\,=\,\frac{60}{12}\;\;\Rightarrow\;\; x\,=\,5y$

The area of the water is: .$\displaystyle \,A\:=\;\frac{1}{2}(5y)(y)\:=\:\frac{5}{2}y^2$

The volume of the water is: . $\displaystyle V\:=\:\frac{5}{2}y^2 \times 25\:=\:\frac{125}{2}y^2$

Differentiate with respect to time: . $\displaystyle \frac{dV}{dt}\:=\:125y\cdot\frac{dy}{dt}$

We are given: $\displaystyle \frac{dV}{dt} = 800$ ft³/min and $\displaystyle y = 5$

So we have: .$\displaystyle 800 \:=\:125(5)\cdot\frac{dy}{dt}$

Therefore: .$\displaystyle \frac{dy}{dt}\:=\:\frac{800}{625}\:=\:1.28$ ft/min.