150 STEP 4. Review the Knowledge You Need to Score High
- Differentiate each term of the equation with respect to time.
- Substitute all known values and known rates of change into the resulting equation.
- Solve the resulting equation for the desired rate of change.
- Write the answer and indicate the units of measure.
Common Related Rate Problems
Example 1
When the area of a square is increasing twice as fast as its diagonals, what is the length of a
side of the square?Letzrepresent the diagonal of the square. The area of a square isA=
z^2
2.
dA
dt
= 2 z
dz
dt(
1
2)
=z
dz
dtSince
dA
dt= 2
dz
dt,2
dz
dt
=z
dz
dt
⇒z= 2.Letsbe a side of the square. Since the diagonalz=2, thens^2 +s^2 =z^2
⇒ 2 s^2 = 4 ⇒s^2 = 4 ⇒s^2 =2ors=√
2.Example 2
Find the surface area of a sphere at the instant when the rate of increase of the volume of
the sphere is nine times the rate of increase of the radius.Volume of a sphere:V=4
3
πr^3 ; Surface area of a sphere:S= 4 πr^2.V=
4
3
πr^3 ;
dV
dt
= 4 r^2
dr
dt.
Since
dV
dt= 9
dr
dt
, you have 9
dr
dt
= 4 πr^2
dr
dt
or 9= 4 πr^2.
SinceS= 4 πr^2 , the surface area isS=9 square units.Note: At 9= 4 πr^2 , you could solve forrand obtainr^2 =9
4 π
orr=3
2
1
√
π. You could then
substituter=3
2
1
√
πinto the formula for surface areaS= 4 πr^2 and obtain 9. These steps
are of course correct but not necessary.Example 3
The height of a right circular cone is always three times the radius. Find the volume of the
cone at the instant when the rate of increase of the volume is twelve times the rate of increase
of the radius.