Expert: Paul Klarreich - 4/23/2007

Question
a balloon in the shape of a cylinder has hemispherical ends of the same radius as that of the cylinder. the balloon is being inflated at the rate of 261pi cubic centimeters per minute. at the instant the radius of the cylinder is 3 centimeters, the volume of the balloon is 144pi cubic centimeters and the radius is increasing at the rate of 2 centimeters per minute. (v of cyl = pi r^2h; v of sphere = 4/3pi r^3)
-at this instant, what is the height of the cylinder?
-at this instant, how fast is the height of the cylinder increasing?

Answer
Questioner:   amy
Category:  Calculus
Private:  No
 
Subject:  related rates
Question:  a balloon in the shape of a cylinder has hemispherical ends of the same radius as that of the cylinder. the balloon is being inflated at the rate of 261 pi cubic centimeters per minute. At the instant the radius of the cylinder is 3 centimeters, the volume of the balloon is 144pi cubic centimeters and the radius is increasing at the rate of 2 centimeters per minute. (v of cyl = pi r^2h; v of sphere = 4/3pi r^3)

-at this instant, what is the height of the cylinder?

-at this instant, how fast is the height of the cylinder increasing?
.............................................................

Hi, Amy,

Your variables and rates here are:

h = height of the cylinder part.
r = radius of the sphere and cylinder.
V = volume of balloon.

dh/dt = rate of increase of height, TO BE FOUND.
dr/dt = rate of increase of radius, given as  2 cm/min.
dV/dt = rate of increase of volume, given as 261 pi cc/min

Assuming that the radius and the 'sphere' have the same radius, your total volume is:

V = pi r^2h + (4/3)pi r^3.

Differentiate:

dV/dt = pi(2rh dr/dt + r^2 dh/dt) + 4 pi r^2 dr/dt

Now we want to put some values in:

r = 3
dr/dt = 2
dV/dt = 261pi

Hmmm -- we now require a value of h, in order to find dh/dt.  So it is now time to use the fact that:

At the instant the radius of the cylinder is 3 centimeters, the volume of the balloon is 144pi

Use the basic relation:  V = pi r^2h + (4/3)pi r^3.

144pi = pi(9)h + (4/3)pi (27)

144pi = 9pi h + 36pi

144 = 9h + 36

16 = h + 4

h = 12  << ANSWER TO FIRST QUESTION.

OK, ready to go:
r = 3;   dr/dt = 2;   dV/dt = 261pi;   h = 12

dV/dt = pi(2rh dr/dt + r^2 dh/dt) + 4 pi r^2 dr/dt

261pi = pi(72(2) + 9 dh/dt) + 4 pi(9)(2)

261pi = pi(144 + 9 dh/dt) + 72pi
189pi = 144pi + 9pi dh/dt
189 = 144 + 9 dh/dt
45 = 9 dh/dt

dh/dt = 5  << ANSWER TO SECOND QUESTION.