A tank in the shape of a right circular cylinder is filled with water (62.5 lb/ft3). It has a height of 8 ft and a diameter of 10 ft. How much work is required to pump all the water to a spout that is 3 ft above the top of the tank?

Easy way: Consider the water to be concentrated at its center of gravity. How much work to lift that mass 4+3 feet?

pi*5^2*8*62.5 lb * 7ft = 274889 ft-lb

Calculus way: consider a thin slice of water of thickness dy. Its weight is

pi * 5^2 * dy * 62.5 = 4908.734 dy lb

Now lift the slice at distance y from the bottom of the tank. It must rise (8-y)+3 = 11-y ft.

So, the work done is the sum of all those works lifting the slices:

integral [0,8] 4908.734(11-y) dy = 274889 ft-lb

Well, that's a tall order! Let's calculate the work required to pump the water out of the tank.

First, we need to find the volume of the water in the tank. The tank is in the shape of a right circular cylinder, so its volume can be calculated using the formula V = π * r^2 * h, where r is the radius and h is the height.

The radius of the tank is half of the diameter, so r = 10 ft / 2 = 5 ft. The height of the tank is 8 ft. Plugging these values into the formula, we get V = π * (5 ft)^2 * 8 ft = 200π ft^3.

Now, let's find the weight of the water. Given that the density of water is 62.5 lb/ft^3, the weight of the water in the tank is W = density * volume = 62.5 lb/ft^3 * 200π ft^3.

To find the work required to pump the water, we need to calculate the vertical distance the water needs to be lifted. The spout is 3 ft above the top of the tank, so the total vertical distance is 8 ft + 3 ft = 11 ft.

The work done is equal to the weight of the water multiplied by the distance it needs to be lifted: Work = weight * distance = 62.5 lb/ft^3 * 200π ft^3 * 11 ft.

Now, if I were a math bot, I would crunch those numbers for you. But as a clown bot, I'll leave it as an exercise for you to calculate the actual work required. Have fun number-crunching! And remember, laughter is the best math teacher!

To find the work required to pump all the water to the spout, we need to calculate the weight of the water in the tank and then multiply it by the distance it needs to be lifted.

Step 1: Find the volume of the tank:
The tank is in the shape of a right circular cylinder, so its volume can be calculated using the formula: V = π * r^2 * h
Where r is the radius (half the diameter) and h is the height.

Given:
Height (h) = 8 ft
Diameter = 10 ft

The radius (r) can be calculated by dividing the diameter by 2:
r = 10 ft / 2 = 5 ft

Plugging in the values:
V = π * (5 ft)^2 * 8 ft
V ≈ 628.318 ft^3 (rounded to 3 decimal places)

Step 2: Find the weight of the water:
The weight of the water can be calculated by multiplying the volume by the density of water.

Given:
Density of water = 62.5 lb/ft^3

Weight = V * density
Weight ≈ 628.318 ft^3 * 62.5 lb/ft^3
Weight ≈ 39,270.875 lb (rounded to 3 decimal places)

Step 3: Find the work required to lift the water:
The work required to lift an object is calculated using the formula: Work = Force * Distance

Given:
Distance = 3 ft (height of the spout above the top of the tank)

Using the weight of the water as the force, we can calculate the work:
Work = Weight * Distance
Work ≈ 39,270.875 lb * 3 ft
Work ≈ 117,812.625 ft-lb (rounded to 3 decimal places)

Therefore, approximately 117,813 ft-lb of work is required to pump all the water to the spout.

To find the work required to pump all the water out of the tank, we first need to calculate the volume of water in the tank, and then convert it to weight. Finally, we can find the work using the formula:

Work = Force x Distance

Step 1: Calculate the Volume of Water
The tank is in the shape of a right circular cylinder, so the volume can be found using the formula:

Volume = π * r^2 * h

Where π is approximately 3.14159, r is the radius of the tank, and h is the height of the tank.

Given that the diameter of the tank is 10 ft, the radius (r) is half of the diameter, which is 10 ft / 2 = 5 ft.

The height (h) of the tank is given as 8 ft.

Thus, the volume of water in the tank can be calculated as:

Volume = π * (5 ft)^2 * 8 ft

Step 2: Convert Volume to Weight
The density of water is given as 62.5 lb/ft^3. To convert volume to weight, we multiply the volume by the density:

Weight = Volume * Density

Step 3: Calculate the Work
Finally, we need to calculate the work required to pump the water to a spout that is 3 ft above the top of the tank. Since the water is being lifted against gravity, the force required can be calculated using:

Force = Weight * Height

Where Height is the height difference between the top of the tank and the spout.

Once we have the force, we can calculate the work using the formula:

Work = Force * Distance

Where Distance is the vertical distance the water needs to be lifted.

Let's calculate each step:

Step 1: Calculate the Volume of Water
Volume = π * (5 ft)^2 * 8 ft ≈ 628.319 ft^3

Step 2: Convert Volume to Weight
Weight = 628.319 ft^3 * 62.5 lb/ft^3 ≈ 39269.93875 lb

Step 3: Calculate the Work
Height = 8 ft + 3 ft = 11 ft

Force = 39269.93875 lb * 11 ft ≈ 431,969.32625 lb·ft

Distance = 11 ft

Work = 431,969.32625 lb·ft * 11 ft ≈ 4,751,662.58875 lb·ft

Therefore, the work required to pump all the water out of the tank to a spout that is 3 ft above the top of the tank is approximately 4,751,662.58875 lb·ft.