Questions: A conical container, oriented such that its vertex is at the bottom, has radius 10 ft and height 40 ft. It is filled to a height of 35 ft of a liquid weighing 64.4 lb / ft^3. How much work will it take to pump the contents to the rim? How much work will it take to pump the liquid to a level of 5 ft above the cone's rim? The amount of work required to pump the liquid to the rim of the tank is [square] ft-lb. (Round to the nearest whole number as needed.)

A conical container, oriented such that its vertex is at the bottom, has radius 10 ft and height 40 ft. It is filled to a height of 35 ft of a liquid weighing 64.4 lb / ft^3. How much work will it take to pump the contents to the rim? How much work will it take to pump the liquid to a level of 5 ft above the cone's rim?

The amount of work required to pump the liquid to the rim of the tank is [square] ft-lb. (Round to the nearest whole number as needed.)

Transcript text: A conical container, oriented such that its vertex is at the bottom, has radius 10 ft and height 40 ft . It is filled to a height of 35 ft of a liquid weighing $64.4 \mathrm{lb} / \mathrm{ft}^{3}$. How much work will it take to pump the contents to the rim? How much work will it take to pump the liquid to a level of 5 ft above the cone's rim? The amount of work required to pump the liquid to the rim of the tank is $\square$ $\mathrm{ft}-\mathrm{lb}$. (Round to the nearest whole number as needed.)

Solution

Solution Steps

To solve this problem, we need to calculate the work required to pump the liquid to the rim of the conical container. The work required to pump a small volume of liquid at a certain height to the rim can be calculated using the formula for work, which involves integrating the force over the distance.

Determine the volume element: Consider a thin horizontal slice of the liquid at height $ y $ with thickness $ dy $. The radius of this slice can be found using similar triangles.
Calculate the volume of the slice: Use the formula for the volume of a cylinder (since the slice is a thin disk).
Calculate the weight of the slice: Multiply the volume of the slice by the weight density of the liquid.
Calculate the work to pump the slice to the rim: Multiply the weight of the slice by the distance it needs to be lifted (which is $ 40 - y $).
Integrate: Integrate this expression from $ y = 0 $ to $ y = 35 $ to find the total work.

Step 1: Determine the Radius of the Slice at Height $ y $

Using similar triangles, the radius $ r_y $ of a thin horizontal slice at height $ y $ is given by: \[ r_y = \frac{10}{40} y = 0.25y \]

Step 2: Calculate the Volume of the Slice

The volume $ dV $ of the thin slice (disk) at height $ y $ with thickness $ dy $ is: \[ dV = \pi r_y^2 dy = \pi (0.25y)^2 dy = 0.0625 \pi y^2 dy \]

Step 3: Calculate the Weight of the Slice

The weight $ dW $ of the thin slice is the volume of the slice multiplied by the weight density of the liquid: \[ dW = 64.4 \times 0.0625 \pi y^2 dy = 4.025 \pi y^2 dy \]

Step 4: Calculate the Work to Pump the Slice to the Rim

The work $ dU $ to pump the thin slice to the rim of the container is the weight of the slice multiplied by the distance it needs to be lifted, which is $ 40 - y $: \[ dU = 4.025 \pi y^2 (40 - y) dy \]

Step 5: Integrate to Find the Total Work

Integrate the expression for $ dU $ from $ y = 0 $ to $ y = 35 $ to find the total work: \[ U = \int_0^{35} 4.025 \pi y^2 (40 - y) dy \]

Step 6: Evaluate the Integral

Evaluating the integral, we get: \[ U = 790954.4271 \pi \] \[ U \approx 2484856.6174 \, \text{ft-lb} \]

Step 7: Round to the Nearest Whole Number

Rounding the total work to the nearest whole number, we get: \[ U \approx 2484857 \, \text{ft-lb} \]