An Olympic-sized swimming pool is a rectangular prism that is 50m long, 25m wide,
�lled with water to a depth of 1.5m. We are going to pump the water out of the pool
with a pump that is 0.5 meters above the surface of the water. Compare the costs
of draining the pool from the surface of the water versus draining the pool from the
bottom. Note that 1 gallon of gasoline (�$4) can do about 36.75 megajoules of work.
The same.
You have to lift the center of mass of the water up to the pump either way. Of course if you attach a tight hose to the outlet you save lifting a half meter by releasing the water at ground level rather than .5 meter up.
total lift if you release water to the atmosphere .5 meter up at the pump = .5 + .75 meters = 1.25 meters
total mass = 1000kg/m^3 * 50*25*1.5 = 1.875 *10^6 kg
total weight = m g = 9.81*1.875*10^6
= 1.84*10^7 Newtons
total work done = weight * distance = 1.84*10^7 *1.25 = 2.23 * 10^7 Joules
2.33 *10^7 /36.75 *10^6 joules/gal
= .634 gal
.634 * 4 $/gal = $2.54
To compare the costs of draining the pool from the surface of the water versus draining the pool from the bottom, we need to calculate the work required in both scenarios.
First, let's determine the volume of water in the pool. The volume of a rectangular prism can be calculated by multiplying its length, width, and depth:
Volume of the pool = Length × Width × Depth
= 50m × 25m × 1.5m
= 1875 cubic meters
Next, let's calculate the energy required to pump the water out of the pool:
Energy = Force × Distance
Draining from the surface:
In this scenario, the pump needs to move the water vertically a distance of 1.5m.
Energy required to pump from the surface = Force × Distance
= Weight of water × Distance
= Volume of water × Density of water × Gravity × Distance
= 1875m^3 × 1000 kg/m^3 × 9.8 m/s^2 × 1.5m
= 27,337,500 Joules
Now, let's calculate the energy required to pump the water out of the pool from the bottom:
In this scenario, the pump needs to move the water vertically a distance of (1.5m + 0.5m) = 2m.
Energy required to pump from the bottom = Force × Distance
= Weight of water × Distance
= Volume of water × Density of water × Gravity × Distance
= 1875m^3 × 1000 kg/m^3 × 9.8 m/s^2 × 2m
= 36,750,000 Joules
Next, let's convert the energy required into gallons of gasoline equivalent:
1 gallon of gasoline (costing $4) can do about 36.75 megajoules of work.
Energy required in both scenarios:
- Surface: 27,337,500 Joules = 0.745 gallon of gasoline equivalent
- Bottom: 36,750,000 Joules = 1 gallon of gasoline equivalent
Therefore, draining the pool from the surface of the water would require 0.745 gallons of gasoline equivalent, while draining it from the bottom would require 1 gallon of gasoline equivalent.
Considering the cost of gasoline, draining the pool from the surface would be more cost-effective.
To compare the costs of draining the pool from the surface of the water versus draining the pool from the bottom, we need to calculate the work required in both scenarios.
1. Draining from the surface:
In this scenario, the pump will need to lift the water from a depth of 1.5 meters to a height of 0.5 meters above the surface. To calculate the work required, we need to determine the weight of the water being lifted.
The weight of the water is given by the formula:
Weight = Volume x Density x Gravitational Acceleration
The volume of water in the pool is the product of the length, width, and depth of the pool:
Volume = Length x Width x Depth
The density of water is approximately 1000 kg/m³.
Using these values, we can calculate the weight of the water in the pool.
2. Draining from the bottom:
In this scenario, the pump will need to lift the water from the bottom of the pool to a height of 0.5 meters above the surface. To calculate the work required, we need to determine the weight of the water being lifted, similar to the previous scenario.
Once we have both values for the work required, we can then compare the costs based on the given information that 1 gallon of gasoline can do about 36.75 megajoules of work and its cost is $4.