Question

An Olympic-size pool is \(50.0 \mathrm{~m}\) long and \(25.0 \mathrm{~m}\) wide. (a) How many gallons of water \((d=1.0 \mathrm{~g} / \mathrm{mL})\) are needed to fill the pool to an average depth of \(4.8 \mathrm{ft} ?\) (b) What is the mass (in kg) of water in the pool?

Short answer

The pool contains approximately 483,296 gallons of water. The mass of the water is 1,828,800 kg.

Step-by-step solution

- Convert pool dimensions to a consistent unit

First, convert the average depth from feet to meters. Use the conversion factor: \[ 1 \text{ft} = 0.3048 \text{m} \] So, \[ 4.8 \text{ft} \times 0.3048 \text{m/ft} = 1.46304 \text{m} \]

- Calculate the volume of the pool

The volume of the pool can be found using the formula for the volume of a rectangular prism: \[ V = \text{length} \times \text{width} \times \text{depth} \] Substitute the values: \[ V = 50.0 \text{m} \times 25.0 \text{m} \times 1.46304 \text{m} = 1,828.8 \text{m}^3 \]

- Convert the volume to liters and gallons

1 cubic meter is equivalent to 1,000 liters. Therefore: \[ 1,828.8 \text{m}^3 \times 1,000 \text{L/m}^3 = 1,828,800 \text{L} \] Using the conversion for liters to gallons: \[ 1 \text{gallon} \approx 3.78541 \text{L} \]So, \[ \frac{1,828,800 \text{L}}{3.78541 \text{L/gallon}} \approx 483,296 \text{gallons} \]

- Calculate the mass of the water

The mass of water can be calculated using the density formula: \[ \text{mass} = \text{density} \times \text{volume} \] Given that the density of water \( d = 1.0 \text{g/mL} \): First, convert the volume from liters to milliliters: \[ 1,828,800 \text{L} \times 1,000 \text{mL/L} = 1,828,800,000 \text{mL} \] So, the mass of water is: \[ \text{mass} = 1.0 \text{g/mL} \times 1,828,800,000 \text{mL} = 1,828,800,000 \text{g} \] Convert the mass to kilograms: \[ \frac{1,828,800,000 \text{g}}{1,000 \text{g/kg}} = 1,828,800 \text{kg} \]

Key concepts

Unit Conversion

Unit conversion is crucial in solving problems involving measurements. In this problem, we need to convert the pool's depth from feet to meters. Feet and meters are common units for measuring length, but they belong to different measurement systems. Using the conversion factor is straightforward. We know: 1 foot = 0.3048 meters. By multiplying the depth in feet by this factor, we get: 4.8 feet × 0.3048 meters/foot = 1.46304 meters. Now all dimensions are in meters, making it easier to work through the next steps.

Volume of a Rectangular Prism

To find the volume of a rectangular prism, we use the formula: \( V = \text{length} \times \text{width} \times \text{depth} \). A rectangular prism is a 3D shape with six faces, all of which are rectangles. The pool is essentially a rectangular prism, and we already have all its dimensions in meters. By plugging in the values: \( V = 50.0 \text{ meters} \times 25.0 \text{ meters} \times 1.46304 \text{ meters} = 1,828.8 \text{ cubic meters} \). This value represents the total volume of water the pool can hold when filled to an average depth of 4.8 feet.

Density Formula

Density is defined as mass per unit volume and is often expressed using the formula: \( \text{density} = \frac{\text{mass}}{\text{volume}} \). In this problem, the density of water is given as 1.0 grams per milliliter (g/mL). To use this value in our calculations, we need to convert the volume into a compatible unit. Since 1 cubic meter equals 1,000 liters and 1 liter equals 1,000 milliliters, we convert: \( 1,828.8 \text{ cubic meters} \times 1,000 \text{ liters/cubic meter} \times 1,000 \text{ milliliters/liter} = 1,828,800,000 \text{ milliliters} \). The mass of the water is then \( 1.0 \text{ grams/milliliter} \times 1,828,800,000 \text{ milliliters} = 1,828,800,000 \text{ grams} \).

Mass Calculation

Finally, to find the mass of the water in kilograms, we need to convert from grams to kilograms. As 1 kilogram equals 1,000 grams, the calculation is: \( \text{mass in kilograms} = \frac{\text{mass in grams}}{1,000} = \frac{1,828,800,000 \text{ grams}}{1,000} = 1,828,800 \text{ kilograms} \). This means the mass of the water needed to fill the pool is 1,828,800 kg. Such large numbers can be daunting, but breaking down each step into manageable conversions helps clarify the solution.