# Related Rates height of cylinder

• ashleyk
In summary, the conversation discusses a balloon in the shape of a cylinder with hemispherical ends, being inflated at a rate of 261 (pi) cubic inches per minute. At an instant when the radius of the cylinder is 3 inches, the volume of the balloon is 144 (pi) cubic inches and the radius is increasing at a rate of 2 inches per minute. The volume of the balloon is calculated using the formulas for the volume of a cylinder and sphere, both of which depend on time. The question also asks for the height of the cylinder and its rate of increase, which can be solved by taking a derivative and finding a relation between the rates and values for the radius and height.
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!

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.

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.

## 1. What is the formula for finding the height of a cylinder?

The formula for finding the height of a cylinder is h = V/πr^2, where h is the height, V is the volume, and r is the radius.

## 2. How do you find the related rate of change for the height of a cylinder?

To find the related rate of change for the height of a cylinder, you would use the formula dV/dt = (πr^2)(dh/dt), where dV/dt is the rate of change of the volume, r is the radius, and dh/dt is the rate of change of the height.

## 3. Can you give an example of a related rates problem involving the height of a cylinder?

One example of a related rates problem involving the height of a cylinder is if you have a cylindrical tank being filled with water at a constant rate, and you need to find the rate at which the height of the water is increasing.

## 4. How does the radius of the cylinder affect the related rate of change for its height?

The radius of the cylinder does not affect the related rate of change for its height. The rate of change of the height is only affected by the rate of change of the volume and the radius remains constant in the formula.

## 5. What units should be used when solving a related rates problem involving the height of a cylinder?

The units used when solving a related rates problem involving the height of a cylinder depends on the given information. The volume and height can have different units, but they must be consistent in the given information and in the formula being used.

• Calculus and Beyond Homework Help
Replies
11
Views
654
• Calculus
Replies
2
Views
2K
• Calculus
Replies
2
Views
2K
• Precalculus Mathematics Homework Help
Replies
5
Views
2K
• Calculus
Replies
6
Views
2K
• Calculus
Replies
1
Views
3K
• Calculus
Replies
2
Views
3K
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus
Replies
4
Views
1K
• Thermodynamics
Replies
5
Views
1K