Problem 48 - Chemistry The Science in Context

Question

A swimming pool is \(7.5 \mathrm{ft}\) deep, 42 ft wide, and \(65 \mathrm{ft}\) long, and it is filled to the brim with water. What is the volume of water in the pool in cubic inches?

Step-by-Step Solution

Answer

Provide your answer in cubic inches. Answer: The volume of the swimming pool is 33,529,920 cubic inches.

Step 1: Convert measurements to inches

Before finding the volume of the swimming pool, we need to convert all given measurements from feet to inches. We know that 1 foot = 12 inches. So, we will convert each measurement by multiplying with 12. Length: \(65 \mathrm{ft} \times 12 = 780 \mathrm{in}\) Width: \(42 \mathrm{ft} \times 12 = 504 \mathrm{in}\) Depth: \(7.5 \mathrm{ft} \times 12 = 90 \mathrm{in}\)

Step 2: Calculate the volume

Now that we have all the measurements in inches, we can calculate the volume of the swimming pool by multiplying the length, width, and depth. Volume = Length × Width × Depth Volume = \(780 \times 504 \times 90\)

Step 3: Find the product

Next, we need to find the product of the three numbers given. \(780 \times 504 \times 90 = 33,529,920\)

Step 4: Write the answer in cubic inches

We have found the volume of the swimming pool in cubic inches: The volume of the swimming pool filled to the brim with water is \(33,529,920 \mathrm{in^3}\).

Key Concepts

Measurement Conversion

Understanding measurement conversion is crucial whenever you're dealing with dimensions that need to be compared or calculated together. In our example, the swimming pool's dimensions were initially given in feet, but to find the volume in cubic inches, we need to perform conversions.

Each dimension—length, width, and depth—must be converted independently from feet to inches. The process is straightforward: since one foot equals 12 inches, we multiply each measurement by 12. For example, the length of the swimming pool is 65 feet, so in inches, it would be:
The process of converting measurements ensures that we have a uniform standard for subsequent calculations—vital for accuracy and understanding in fields ranging from construction to baking.

Dimensional Analysis

Dimensional analysis is the method by which you check that your conversion of units makes sense within the context of a calculation. It's like a road map for ensuring that you end up with the correct units for your final answer, especially when multiple conversions are involved. In this context, you want to calculate the volume of the swimming pool in cubic inches.

After converting the individual linear dimensions to inches using measurement conversion, we then use dimensional analysis to ensure that when we multiply the length, width, and depth together, we indeed get a volume measurement. Since volume is a three-dimensional space, it's represented in cubic units, such as cubic inches. The formula we use is: The result of our length, width, and depth multiplication then confirms that our volume will indeed be in the correct cubic unit.

Unit Conversion

Finally, we have unit conversion, which is the process of converting one type of unit into another within the same measurement system. In the context of our swimming pool problem, we converted feet to inches, which are both units of length within the Imperial system. It's imperative to use a correct conversion factor, which for feet to inches is 1 foot = 12 inches.

The importance of unit conversion can't be overstated. Imagine if you accidentally used a conversion for converting feet to meters, such as 1 foot = 0.3048 meters, in our pool problem. The final volume would end up in cubic meters instead of cubic inches, rendering the solution incorrect for the given context. This highlights the necessity of attention to detail and the proper application of unit conversion factors. The ability to correctly convert units underpins the accuracy of measurements in science, industry, and daily life, ensuring that everyone involved in a task or project is literally 'on the same page'.