Related Rates height of cylinder

[ashleyk]

A balloon is in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder. The balloon is being inflated at the rate of 261 (pi) cubic inches per minute. At the instant the radius of the cylinder is 3 inches, the volume of the balloon is 144 (pi) cubic inches and the radius of the cylinder is increasing at the rate of 2 inches per minute. (Using the formulas for the volume of a cylinder= (pi)(r^2)h and the formula for the volume of a sphere= (4/3)(pi)(r^3) )
A. At the instant, what is the height of the cylinder?
B. At this instant, how fast is the height of the cylinder increasing?

I found part A to be 16/(pi) but I don't know if that is right. I don't know where to go for part B. I know I have to take a derivative somewhere but I'm lost. Any help would be great, this is due tomorrow for a grade...and I NEED THE HELP! Thanks!

 

[dextercioby]

1.The (2-)sphere is a just a surface & it has zero volume...
2.The cylinder is just a surface and it has zero volume.
2.The volume of the air/gas inside the balloon is

$$V(r,h)=\frac{4\pi r^{3}}{3}+\pi r^{2} h$$

All functions in the above formula depend on time...

Diff.wrt time & get a relation between rates & values for "r,"h".

Daniel.

 

[M98Ranger]

A. To find the height of the cylinder, we can use the formula for the volume of a cylinder: V = (pi)(r^2)h. We are given that the volume is 144 (pi) cubic inches and the radius is 3 inches. Plugging in these values, we get:

144 (pi) = (pi)(3^2)h
144 = 9h
h = 16 inches

Therefore, at the instant when the radius is 3 inches, the height of the cylinder is 16 inches.

B. To find the rate at which the height is increasing, we can use the related rates formula:

dV/dt = (pi)(2r)(dr/dt) + (4/3)(pi)(r^2)(dh/dt)

We know that dV/dt = 261 (pi) cubic inches per minute, r = 3 inches, and dr/dt = 2 inches per minute. Plugging in these values, we get:

261 (pi) = (pi)(2)(3)(2) + (4/3)(pi)(3^2)(dh/dt)
261 = 12pi + 12pi(dh/dt)
261 = 24pi + 12pi(dh/dt)
dh/dt = (261 - 24pi)/(12pi) = 21.5 inches per minute

Therefore, at the instant when the radius is 3 inches, the height of the cylinder is increasing at a rate of 21.5 inches per minute.