A tank is full of water. Find the work required to pump the water out of the spout. Use the fact that water weighs 62.5 lb/ft3. (Assume r = 4 ft, R = 8 ft, and h = 15 ft.)
v = pi r^2 h
the center of mass is at h=15/2
so the work required is volume * density * 62.5 * 15/2
To find the work required to pump the water out of the spout, we need to calculate the weight of the water and then multiply it by the height it needs to be lifted.
First, let's calculate the volume of water in the tank. The tank is in the shape of a frustum of a cone, so we can use the formula for the volume of a frustum of a cone:
V = (1/3)πh(R^2 + r^2 + Rr)
Given:
r = 4 ft (radius of the smaller circular base)
R = 8 ft (radius of the larger circular base)
h = 15 ft (height of the frustum)
Plugging in the values, we have:
V = (1/3)π(15)((8^2) + (4^2) + (8)(4))
= (1/3)π(15)(64 + 16 + 32)
= (1/3)π(15)(112)
= 560π ft^3
Now, let's calculate the weight of the water:
Weight = Volume × Density
Given:
Density of water = 62.5 lb/ft^3
Weight = 560π ft^3 × 62.5 lb/ft^3
≈ 34,906 lb
Finally, let's calculate the work required to pump the water out:
Work = Weight × Height
Given:
Height = 15 ft
Work = 34,906 lb × 15 ft
= 523,590 ft-lb
Therefore, the work required to pump the water out of the spout is approximately 523,590 ft-lb.
To find the work required to pump the water out of the spout, we need to calculate the weight of the water in the tank first. The weight of an object can be calculated using the formula:
Weight = Mass × Acceleration due to gravity
Since density = mass/volume, for water we have:
Density of water = Mass of water/Volume of water
Rearranging the formula, we get:
Mass of water = Density of water × Volume of water
Now, let's calculate the volume of water in the tank. The shape of the tank is in the form of a frustum of a cone. The volume of a frustum of a cone can be calculated using the formula:
Volume = 1/3 × π × h × (R² + R × r + r²)
Given the values for r = 4 ft, R = 8 ft, and h = 15 ft, we can substitute these values into the formula to find the volume of water in the tank.
Volume = 1/3 × π × 15 ft × (8 ft² + 8 ft × 4 ft + 4 ft²)
After calculating the volume, we can find the weight of the water by multiplying the density of water (62.5 lb/ft³) by the volume of water.
Weight = Density of water × Volume
Finally, the work required to pump the water out of the spout can be found using the formula:
Work = Force × Distance
In this case, the force is the weight of the water and the distance is the height of the tank (15 ft). Therefore, we can calculate the work required by multiplying the weight by the height of the tank.
Work = Weight × Height
Well, here's a riddle for you: Why did the tank want to be emptied? Because it heard there was a spout party and it wanted to make a splash!
Now, let's get serious and do some math. To find the work required to pump the water out of the spout, we need to calculate the weight of the water in the tank. The weight of an object can be found by multiplying its mass by the acceleration due to gravity. In this case, we can find the weight of the water by multiplying its volume by its density (which is given as 62.5 lb/ft^3).
The volume of the water in the tank can be calculated using the formula for the volume of a frustum of a cone, which is given by V = (1/3) * π * h * (r^2 + R^2 + r * R), where h is the height of the frustum, r is the radius of the smaller base, and R is the radius of the larger base.
Substituting the given values, we have V = (1/3) * π * 15 * (4^2 + 8^2 + 4 * 8).
Now, we can plug in the values and calculate the volume of the water in the tank.
V = (1/3) * π * 15 * (16 + 64 + 32)
= (1/3) * π * 15 * 112
≈ 1659.77 ft^3
The weight of the water can be calculated by multiplying the volume by the density.
Weight = V * density
= 1659.77 * 62.5
≈ 103735.63 lb
So, the work required to pump the water out of the spout is approximately 103735.63 lb.
I hope this answer didn't make you feel all washed up!