Problem 30 - Basic Technical Mathematics with Calculus

Question

\(\text {Solve the given problems.}\) A swimming pool is \(50.0 \mathrm{ft}\) wide, \(78.0 \mathrm{ft}\) long. \(3.50 \mathrm{ft}\) deep at one end, and \(8.75 \mathrm{ft}\) deep at the other end. How many cubic feet of water can it hold? (The slope on the bottom is constant.) See Fig. 2.122

Step-by-Step Solution

Answer

The swimming pool can hold 23,887.5 cubic feet of water.

Step 1: Determine the average depth

Since the swimming pool has a constant slope from one end to the other, calculate the average depth by taking the average of the depths at both ends. The average depth is: \[\frac{3.50 + 8.75}{2} = \frac{12.25}{2} = 6.125 \text{ ft}\]

Step 2: Calculate the area of the base

The base of the swimming pool is a rectangle with a width of 50.0 ft and a length of 78.0 ft. Find the area by multiplying these two dimensions:\[50.0 \times 78.0 = 3900 \text{ square feet}\]

Step 3: Determine the volume of the pool

Multiply the area of the base by the average depth to find the volume of the swimming pool:\[3900 \times 6.125 = 23887.5 \text{ cubic feet}\]

Key Concepts

Volume Calculation

Calculating the volume of a swimming pool is essential to determine how much water it can hold. Volume calculation in geometry typically involves multiplying the area of the base by the height (or depth, in the case of pools). For a rectangular pool:

• Calculate the area of the rectangular base.
• Determine the average depth, if the pool bottom is sloped.
• Multiply the area by the average depth.

You'll find the total volume in cubic units, like cubic feet. Knowing the volume helps in various practical tasks, such as maintaining water levels or adding chemicals efficiently. Volume calculation is a universal principle, useful in numerous scenarios beyond pools, such as calculating space in storerooms or containers.

Swimming Pool Dimensions

Swimming pool dimensions are key in determining not just the aesthetics of your pool, but its functionality as well. The dimensions of a pool include the width, length, and the depth at various points:

• Width and length are straightforward measures of the pool's surface area.
• Depth may vary, especially with sloped bottoms, which is common in pools.

The surface area is crucial when considering evaporation rates or the amount of materials needed for maintenance. Accurately measuring the pool dimensions helps in optimizing design and ensuring safety standards are met. For our pool, a width of 50 ft and length of 78 ft makes the base a sizeable rectangle, ideal for calculations.

Average Depth Calculation

The average depth calculation is necessary when dealing with pools that have a sloped bottom. This ensures that calculations regarding volume are accurately portrayed. To find the average depth:

• Add together the deepest point and the shallowest point of the pool.
• Divide by two.

This provides the average depth, which accurately reflects the pool's overall depth profile for calculations. In our exercise, the pool is 3.50 ft at the shallow end and 8.75 ft at the deep end. Thus, the average depth was found to be 6.125 ft. Accurate average depth calculation is vital to ensure that volume measurements are precise and reliable. This average helps in managing water treatment and ensuring proper safety levels.