Problem 47 - Thinking Mathematically

Question

A circular backyard pool has a diameter of 24 feet and is 4 feet deep. One cubic foot of water has a capacity of approximately \(7.48\) gallons. If water costs \(\$ 2\) per thousand gallons, how much, to the nearest dollar, will it cost to fill the pool?

Step-by-Step Solution

Answer

The cost to fill the pool to the nearest dollar can be obtained by and following through with these calculations.

Step 1: Calculate the Volume of the Pool

The pool is circular, so the formula to calculate the volume of a cylinder (V) is given by \(V = \pi r^2 h\) where \(r\) is the radius and \(h\) is the height. The diameter of the pool is 24 feet, so its radius is \(r = \frac{24}{2} = 12\) feet. The depth or height of the pool is \(h = 4\) feet. Substituting these values into the formula, we get: \(V = \pi \cdot 12^2 \cdot 4\)

Step 2: Convert the Volume to Gallons

Given that 1 cubic foot of water is approximately 7.48 gallons, we can convert the volume of the pool to gallons by multiplying the volume in cubic feet by 7.48. Thus, the volume in gallons is \(V' = V \cdot 7.48\).

Step 3: Determine the Cost to Fill the Pool

Given that the cost of water is $2 per thousand gallons, we can calculate the cost to fill the pool by multiplying the volume in thousands of gallons by the cost per thousand gallons. The Volume in thousands of gallons is obtained by dividing the volume in gallons (V') by 1000 (V''=V'/1000). The cost to fill the pool (C) is then given by \(C = 2 \cdot V''\)

Key Concepts

Cubic Feet to Gallons Conversion

When working with large volumes of water, such as filling a pool, we often need to convert between different units of volume. In this case, we need to convert cubic feet to gallons. It is important to remember that 1 cubic foot of water is approximately equivalent to 7.48 gallons.
To convert from cubic feet to gallons, simply multiply the volume in cubic feet by 7.48.

• If your pool has a volume of 100 cubic feet, the calculation to find the volume in gallons would be: \[ 100 \times 7.48 = 748 \text{ gallons} \]

Understanding this conversion is crucial for estimating how much water is needed, especially when considering cost and water supply limitations.
Keep this conversion factor in mind whenever you're dealing with water-related calculations.

Cost Calculation

Once you know the volume of water in gallons, the next step is to calculate the cost to fill the pool. The typical way to determine this is by knowing the cost per unit volume of water, usually per thousand gallons.
For example, if water costs $2 per thousand gallons, you would first calculate your water volume in thousands of gallons.

• Say your pool holds 10,000 gallons, you would convert this to thousands by dividing by 1,000: \[ \frac{10,000}{1,000} = 10 \text{ thousand gallons} \]
• Then, multiply this result by the cost per thousand gallons: \[ 10 \times 2 = 20 \text{ dollars} \]

This way, you accurately estimate the cost, helping you budget for the expense of filling your pool.

Circular Pool Dimensions

Circular pool dimensions are often given in terms of diameter and depth, and it's essential to know how to use these measurements to find the pool's volume. The volume is calculated using the formula for the volume of a cylinder, which applies to circular pools:\[ V = \pi r^2 h \]

• Diameter to Radius: Since the formula requires radius, convert the diameter by dividing it by 2. For example, a 24-foot diameter becomes a 12-foot radius.
• Using the Formula: With a radius of 12 feet and a depth (or height) of 4 feet, plug these values in: \[ \pi \times 12^2 \times 4 = \pi \times 144 \times 4 = 576\pi \text{ cubic feet} \]

Understanding the relationship between diameter, radius, and height, along with the cylinder volume formula, enables you to calculate volumes for any circular pool configuration.