A pool is in the shape of a rectangular prism with a length of 15 feet, a width of 10 feet, and a depth of 4 feet. The pool is filled with water at a rate no faster than 3 cubic feet per minute. How many minutes t does it take to fill the pool?
Answers
From the given information we can calculate the ratio
d
h
d
V
where
h
is the height of water in the deep end and
V
is the volume of water.
We are given
d
V
d
t
and multiplying the two ratios together we can calculate
d
h
d
t
enter image source hereThe Volume of water in the pool is given by the formula
V
=
w
×
l
×
h
2
where
h
is the height of water at its deepest point,
l
is the length of the surface area of the water, and
w
is the (constant) width of the water surface.
w
=
25
l
h
=
40
6
provided
h
≤
6
(by similar triangles)
→
l
=
20
3
h
and the formula for the Volume of water can be rewritten as
V
(
h
)
=
25
×
20
3
h
×
h
2
or
V
(
h
)
=
250
h
2
3
d
V
d
h
=
500
h
3
→
d
h
d
V
=
3
500
h
The rate of change in depth of the water per unit of time is
d
h
d
t
=
d
h
d
V
⋅
d
V
d
t
We are told
d
V
d
t
=
10
cu.ft./min.
So when the water is 4 feet deep (unfortunately I labelled this
h
for height), the rate of change in depth (aka height) of the water is
d
h
d
t
=
3
500
⋅
4
⋅
10
=
3
200
=
0.015
ft/min.