You are visiting your friend Fabio’s house. You find that, as a joke, he filled his swimming pool with Kool-Aid, which dissolved perfectly into the water. However, now that you want to swim, you must remove all of the Kool-Aid contaminated water. The swimming pool is round, with a 15.5 foot radius. It is 8 feet tall and has 5 feet of water in it. How much work is required to remove all of the water by pumping it over the side? Use the physical definition of work, and the fact that the density of the Kool-Aid contaminated water is σ= 64.1lbs/ft3
1st way: figure the weight of a slice of water in the pool.
Integrate that weight through a height of its distance from the top.
2nd way: Figure the weight of all the water in the pool. Its center of mass is 5.5 feet from the top of the pool, so lift that weight 5.5 feet.
To calculate the work required to remove all of the water from the swimming pool, we can use the physical definition of work:
Work = force × distance
In this case, the force can be calculated by finding the weight of the water to be removed. The weight of an object is given by the equation:
Weight = mass × acceleration due to gravity
The mass of the water can be found using its volume and density:
Mass = density × volume
The volume of water in the swimming pool can be calculated using the formula for the volume of a cylinder:
Volume = π × radius^2 × height
Now let's go through the calculations step-by-step:
Step 1: Calculate the volume of the water in the swimming pool.
Volume = π × radius^2 × height
Volume = π × (15.5 ft)^2 × 5 ft
Volume ≈ 11961.846 ft^3
Step 2: Calculate the mass of the water.
Mass = density × volume
Mass = 64.1 lbs/ft^3 × 11961.846 ft^3
Mass ≈ 766,402.8826 lbs
Step 3: Calculate the weight of the water.
Weight = mass × acceleration due to gravity
Weight = 766,402.8826 lbs × 32.2 ft/s^2
Weight ≈ 24,684,290.50212 ft·lbs
Step 4: Calculate the work required.
Work = force × distance
In this case, the distance is the height of the pool, which is 8 ft.
Work = weight × height
Work = 24,684,290.50212 ft·lbs × 8 ft
Work ≈ 197,474,324.01696 ft·lbs
Therefore, approximately 197,474,324.01696 ft·lbs of work is required to remove all of the Kool-Aid contaminated water from the swimming pool by pumping it over the side.
To calculate the work required to remove all of the water from the swimming pool, we need to understand the physical definition of work and assume that the entire pool is filled with water.
The physical definition of work is given by the equation:
Work = Force x Distance
In this case, the force required to remove the water is equal to the weight of the water, which can be calculated using the density and volume of the water.
The volume of water in the pool can be found using the formula for the volume of a cylinder:
Volume = π * r^2 * h
where r is the radius of the pool and h is the height of the water.
Using the given values:
Radius (r) = 15.5 feet
Height of water (h) = 5 feet
Substituting these values into the volume formula:
Volume = π * (15.5^2) * 5
Next, we can calculate the weight of the water using the formula:
Weight = Density x Volume x Gravity
where:
Density (ρ) = 64.1 lbs/ft^3 (given)
Gravity (g) = 32.2 ft/s^2 (acceleration due to gravity)
Substituting the values into the formula:
Weight = 64.1 lbs/ft^3 * (π * (15.5^2) * 5) * 32.2 ft/s^2
Finally, we can calculate the work required to remove the water by multiplying the weight by the distance that the water needs to be lifted (which is equal to the height of the pool):
Work = Weight x Height of water
Substituting the values:
Work = [64.1 lbs/ft^3 * (π * (15.5^2) * 5) * 32.2 ft/s^2] * 8 ft
Simplifying the equation and calculating the result will give us the work required to remove all of the water from the pool.