# Work done against gravity on a right circular cone

• Onodeyja
In summary, a right circular cone with vertex down and a height of 10 feet and base radius of 5 feet is filled with a fluid with varying density that changes linearly with distance to the top. The density varies according to the equation 120 - 4y, where y is the height. The work needed to pump out all the fluid to the top of the cone can be calculated by integrating the force, which is equal to the mass (found by multiplying the volume of a layer of the solid by the density) multiplied by gravity, over the height of the cone. The radius of the cone changes as the height increases, and can be found by considering the slanted side of the cone in the xy plane, with coordinates
Onodeyja

## Homework Statement

A right circular cone has vertex down and is 10 feet tall with base radius 5 feet. The cone is filled with a fluid having varying density. The density varies linearly with distance to the top. Here "varies linearly" means the quantities are related by an equation of at most degree 1. At the top of the cone, the density is 80 lbs/ft^3, and at the bottom the density is 120 lbs/ft^3. How much work in ft-lbs is needed to pump out all the fluid to the top of the cone.

## The Attempt at a Solution

Well, I figured out that the equation for the density would be 120 - 4y (taking 120 - 80, then dividing that answer by the height gives 4). The total work will be found by integrating from 0 to 10. From physics, I know that work is equal to force times distance. In order to get the force you need to know the mass (multiplied by gravity). The mass is found by multiplying the volume of a layer of the solid by the density.

I'm having trouble finding the area at any height of the cone. I know that the radius changes as the height increases eventually getting to 5 feet.

m = pV = (120-4y)(π(r(y)^2))
F = 9.8(120-4y)(π(r(y)^2))
W = ∫[0,10] 9.8(120-4y)(π(r(y)^2))y dy

Last edited:
Look at the slanted side of the cone; say it's in the xy plane. It goes from (0,0) to (5,10). What is the equation of that line? The radius you want is the x on that line in terms of y.

## 1. What is the formula for calculating work done against gravity on a right circular cone?

The formula for calculating work done against gravity on a right circular cone is W = mgh, where W is the work done, m is the mass of the object, g is the acceleration due to gravity, and h is the height of the cone.

## 2. How does the angle of the cone affect the work done against gravity?

The angle of the cone does not affect the work done against gravity. As long as the height and mass of the object remain constant, the work done will be the same regardless of the angle of the cone.

## 3. Can negative work be done against gravity on a right circular cone?

No, negative work cannot be done against gravity on a right circular cone. The object must be lifted against the force of gravity, so the work done will always be positive.

## 4. Is the work done against gravity on a right circular cone dependent on the radius of the base?

No, the work done against gravity is not dependent on the radius of the base. As long as the height and mass of the object remain constant, the work done will be the same regardless of the radius of the base.

## 5. How can this concept be applied in real-life scenarios?

The concept of work done against gravity on a right circular cone can be applied in various real-life scenarios, such as calculating the work done when lifting an object to the top of a cone-shaped structure, or determining the energy required to move an object up a cone-shaped ramp.

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