1. When air is blown into a football, its radius is tripled. The ratio of the surface area of the original football to the inflated one is
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Answer:
This is a Related Rates (of change) problem.
The rate at which air is being blown in will be measured in volume per unit of time. That is a rate of change of volume with respect to time. The rate at which air is being blown in is the same as the rate at which the volume of the balloon is increasing.
V
=
4
3
π
r
3
We know
d
r
d
t
=
5
cm/sec
. We want
d
V
d
t
when
r
=
13
cm
.
Differentiate
V
=
4
3
π
r
3
implicitly with respect to
t
d
d
t
(
V
)
=
d
d
t
(
4
3
π
r
3
)
d
V
d
t
=
4
3
π
⋅
3
r
2
d
r
d
t
=
4
π
r
2
d
r
d
t
Plug in what you know and solve for what you don't know.
d
V
d
t
=
4
π
(
13
cm
)
2
(
5
cm/sec
)
=
20
⋅
169
⋅
π
cm
3
/sec
The air is being blown in at a rate of
3380
π
cm
3
/sec
.
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