# Cylinder: Ratio Volume/Surface Area

• LURCH
In summary, a person was trying to determine the cooling rates for a pot of coffee by changing the fluid level. They believed that the ratio of surface area to volume would be the main factor in determining the cooling rate. They asked if anyone knew of a formula that could calculate this ratio for a cylinder with a constant radius and varying height. They also shared their thoughts on the relationship between volume and surface area for 3-D objects. Finally, they thanked anyone in advance for any help they could provide.
LURCH
So a friend of mine was trying to figure out cooling rates for a pot of coffee. He wanted to see how much faster the coffee would cool when he doesn't fill the pot all the way. He is starting from the assumtion that the main determinant for cooling rate will be how the ratio of Surface Area to Volume changes as the fluid level changes. Since the fluid essentially forms a cylinder, and the height is the only real variable dimension, I thought there must be a formula that will yield this ratio, because volume is a function of surface area.

Does anyone know of such a formula? One that will yield the ratio of Volume to Surface Area of a cylinder when the radius remains constant and the height varies?

My thoughts so far:
If one dimension of any 3-D object is increased, the surface area of that object will increase by the square of that dimension, but the volume will increase by the cube of the dimension, right? So won't the volume always be the surface area to the power of three-halves (v=a3/2)?

Since the volume and surface area are both functions of height, V(h) = pi*r^2*h and S(h) = 2Pi*r(h + r) imply that R(h) = V(h)/S(h) = r*h/[2(r + h)]. r is treated as a constant.

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

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

## 2. How do you calculate the surface area of a cylinder?

The surface area of a cylinder can be calculated using the formula SA = 2πr^2 + 2πrh, where r is the radius and h is the height.

## 3. How does the ratio of volume to surface area of a cylinder affect its shape?

The higher the ratio of volume to surface area, the more "fat" or "stout" the cylinder will appear. This is because a larger volume relative to the surface area means the cylinder has a larger diameter and a shorter height.

## 4. Can you explain why the ratio of volume to surface area of a cylinder is important?

The ratio of volume to surface area of a cylinder is important in various fields of science, such as chemistry and physics. In chemistry, the ratio can affect the rate of reaction and the efficiency of diffusion. In physics, the ratio can affect the buoyancy and heat transfer of objects.

## 5. How does the ratio of volume to surface area of a cylinder change when the height is doubled?

If the height of a cylinder is doubled, the ratio of volume to surface area will decrease. This is because the volume will increase by a factor of 2^2 (4 times), while the surface area will only increase by a factor of 2^1 (2 times).

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