Problem 27 - Calculus Early Transcendentals

Question

A swimming pool has the shape of a box with a base that measures \(25 \mathrm{m}\) by \(15 \mathrm{m}\) and a uniform depth of \(2.5 \mathrm{m}\). How much work is required to pump the water out of the pool when it is full?

Step-by-Step Solution

Answer

Answer: The work required to pump all the water out of the pool is 11,496,093.75 Joules.

Step 1: Calculate the volume of water in the pool

First, calculate the total volume of water in the pool using the formula for the volume of a rectangular prism: Volume = length × width × depth Volume = 25m × 15m × 2.5m Volume = 937.5 m³

Step 2: Calculate the mass of the water

After obtaining the volume of water, you can now find the mass of the water by using the following formula: Mass = Density × Volume Considering that the density of water is 1000 kg/m³, you have: Mass = 1000 kg/m³ × 937.5 m³ Mass = 937500 kg

Step 3: Find the center of mass of the water

In order to find the work required to pump out the water, we need to find the height through which the center of mass of the water must be lifted. Since the pool has a uniform depth, the center of mass of the water is at half the depth. So, in this case: Center of mass height = 2.5m / 2 Center of mass height = 1.25m

Step 4: Calculate the force required to lift the water

Now, we need to calculate the gravitational force acting on the mass of water, using the following formula: Force = Mass × Gravitational acceleration Considering that the gravitational acceleration is approximately 9.81 m/s², we have: Force = 937500 kg × 9.81 m/s² Force = 9196875 N (Newtons)

Step 5: Calculate the work required to pump the water out

Finally, we can calculate the work required to pump the water out of the pool by using the following formula: Work = Force × Distance × cos(angle) Since we are lifting the water vertically, the angle between the force and the displacement is 0 degrees, so cos(angle) is equal to 1, and the formula simplifies to: Work = Force × Distance Work = 9196875 N × 1.25 m Work = 11496093.75 J (Joules) Therefore, the work required to pump the water out of the pool when it is full is 11,496,093.75 Joules.

Key Concepts

Volume of a Rectangular Prism

To understand how much water is in a swimming pool shaped like a rectangular box, we start by calculating the pool's volume. The pool's volume tells us how much space it occupies, which is crucial for figuring out how much work is needed to pump it empty. A rectangular prism is a 3D object like a box or a pool. To find its volume, use the formula:\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Depth} \]In the pool's case:- Length = 25 meters- Width = 15 meters- Depth = 2.5 metersPlug these numbers into the formula to get the volume:\[ \text{Volume} = 25 \times 15 \times 2.5 = 937.5 \text{ cubic meters} \]The volume of the pool tells us how much water is inside. This value plays a key role in the next steps involving density and mass calculations.

Density of Water

Density is a measure of how much mass is contained in a given volume. For water, this concept is fairly standard and fixed. Understanding water's density helps us find out how heavy the water in the pool is. Water's density is usually accepted as:- \(1000 \text{ kilograms per cubic meter} (kg/m^3)\)Once you have the volume of the water, calculating the mass is straightforward. Multiply the volume by the density. For our pool:- Volume = 937.5 cubic metersThus:\[ \text{Mass} = \text{Density} \times \text{Volume} = 1000 \times 937.5 = 937500 \text{ kilograms} \]This means all the water in the pool weighs 937,500 kilograms. The mass of the water is essential for calculating the force needed to move it.

Center of Mass

When you're dealing with volumes of substances, such as the water in a pool, knowing where the center of mass is becomes valuable. It's the point at which the mass of the body or object is concentrated, and it behaves as though all its mass is centered at that point.For a uniform rectangular prism like our swimming pool, the center of mass is halfway up the depth. This is because the pool's water is uniformly distributed.In the pool's case:- Total depth = 2.5 metersThe center of mass height:\[ \text{Center of Mass Height} = \frac{2.5}{2} = 1.25 \text{ meters} \]Knowing this height allows us to calculate the work needed to lift the water out of the pool against the force of gravity.

Gravitational Force

Understanding gravitational force is key to calculating the work needed to pump the pool dry. Gravity is the force pulling objects towards Earth's center, and we use it here to determine the force required to lift the water out.The force of gravity on an object is calculated using the object's mass and the gravitational acceleration:- Gravitational acceleration = \(9.81 \text{ m/s}^2\)The formula to find force is:\[ \text{Force} = \text{Mass} \times \text{Gravitational Acceleration} \]For our pool water:- Mass = 937,500 kilograms\[ \text{Force} = 937,500 \times 9.81 = 9,196,875 \text{ Newtons} \]This force reveals the effort needed to lift the water. Finally, to find the work done, multiply this force by the distance the center of mass moves (1.25 meters in this case), resulting in 11,496,093.75 Joules to empty the pool completely.