The volume of an Olympic swimming pool is 6.6043 x 10 gallons. If the length of the pool is 50.00 m and the width is 25.00 m, how deep is an Olympic sized pool? a. What is the equation for the volume of a box? b. What SI unit does volume have? c. To figure out the depth, we need to convert gallons to the SI unit! This can be done using your metric prefixes and the two conversion factors listed below. 1 ml = 1 cm (exact conversion) IL -0.264172 gallons What is the volume of an Olympic pool in SI units? How many sig figs should this number have? d. Use your answer from parte and the equation from part a to determine the depth of the pool? Report your answer in meters.
The Correct Answer and Explanation is :
Let’s break this down step by step:
a. Equation for the volume of a box
The volume ( V ) of a rectangular box (such as a pool) is calculated using the formula:
[
V = \text{length} \times \text{width} \times \text{depth}
]
In this case, we know the length and width of the Olympic pool, and we need to determine the depth.
b. SI unit of volume
The SI unit for volume is the cubic meter (m³). Since we are converting from gallons to cubic meters, we will first need to convert gallons into a volume unit compatible with the SI system.
c. Converting the volume of the Olympic pool to SI units
We are given the volume of the pool in gallons:
[
6.6043 \times 10^4 \text{ gallons}
]
We need to convert gallons to liters (L), and then from liters to cubic meters (m³).
Conversion factors:
1 liter (L) = 1,000 milliliters (mL)
1 milliliter (mL) = 1 cubic centimeter (cm³)
1 gallon = 3.78541 liters (L)
To convert 6.6043 × 10⁴ gallons to liters:
[
6.6043 \times 10^4 \text{ gallons} \times 3.78541 \text{ L/gallon} = 2.5006 \times 10^5 \text{ L}
]
Now, convert liters to cubic meters:
[
2.5006 \times 10^5 \text{ L} \times \frac{1 \text{ m}^3}{1,000 \text{ L}} = 250.06 \text{ m}^3
]
Thus, the volume of the pool is approximately 250.06 m³.
Since we started with 5 significant figures in the original volume (6.6043 × 10⁴), we should keep 5 significant figures in the converted volume. Therefore, the volume of the pool is 250.06 m³.
d. Determining the depth of the pool
We now use the volume formula for a box to find the depth of the pool. The length is 50.00 meters, and the width is 25.00 meters. Rearranging the formula to solve for depth:
[
\text{depth} = \frac{V}{\text{length} \times \text{width}} = \frac{250.06 \text{ m}^3}{50.00 \text{ m} \times 25.00 \text{ m}} = \frac{250.06}{1250} = 0.200 \text{ m}
]
Thus, the depth of the pool is 0.200 meters.
Explanation:
- Volume Equation: The volume of a pool is calculated by multiplying its length, width, and depth.
- Unit Conversion: We converted gallons to liters using the conversion factor 1 gallon = 3.78541 liters and then liters to cubic meters, which is the SI unit for volume.
- Significant Figures: We ensured the result maintained the appropriate number of significant figures, rounding to 5 significant digits.
- Depth Calculation: By rearranging the volume equation, we were able to solve for the depth and find it to be 0.200 meters.
This gives us the depth of an Olympic pool in the SI unit of meters.