Problem 70 - College Physics

Question

Approximately 118 mi wide, 307 mi long, and averaging \(279 \mathrm{ft}\) in depth, Lake Michigan is the secondlargest Great Lake by volume. Estimate its volume of water in cubic meters.

Step-by-Step Solution

Answer

The estimated volume of Lake Michigan is approximately \(7.96 \times 10^{12}\) cubic meters.

Step 1: Convert Length Dimensions to Meters

First, we need to convert the measurements of Lake Michigan from miles and feet to meters. The dimensions given are: - Width: 118 miles - Length: 307 miles - Average Depth: 279 feet To convert miles to meters, use the conversion factor: 1 mile = 1609.34 meters. Therefore: - Width in meters = 118 miles * 1609.34 m/mile = 189901.32 meters - Length in meters = 307 miles * 1609.34 m/mile = 494067.38 meters.

Step 2: Convert Depth to Meters

Next, convert the average depth from feet to meters using the conversion factor: 1 foot = 0.3048 meters. Therefore: - Average depth in meters = 279 feet * 0.3048 m/foot = 85.0488 meters.

Step 3: Calculate the Volume in Cubic Meters

Now, calculate the volume of Lake Michigan in cubic meters using the formula for volume of a rectangular prism: \[ \text{Volume} = \text{Width} \times \text{Length} \times \text{Depth} \]Substituting the converted dimensions:\[ \text{Volume} = 189901.32 \text{ m} \times 494067.38 \text{ m} \times 85.0488 \text{ m} \]\[ \text{Volume} \approx 7.9619 \times 10^{12} \text{ cubic meters} \]

Step 4: Conclusion

Therefore, the estimated volume of Lake Michigan is approximately \(7.96 \times 10^{12}\) cubic meters. This large number demonstrates the enormous volume capacity of Lake Michigan.

Key Concepts

Unit Conversion

In many math and science problems, converting units is a crucial skill. It allows you to work within a consistent measurement system, which simplifies calculations and provides clearer understanding. Converting units involves multiplying by a conversion factor—a known ratio that helps bridge one unit of measure to another.
For instance, when converting miles to meters, the conversion factor is 1 mile equals 1609.34 meters. So, to find out how many meters 118 miles represent, you multiply:

• 118 miles \( \times \) 1609.34 meters/mile = 189901.32 meters.

Similarly, converting depth from feet to meters involves the factor 1 foot equals 0.3048 meters:

• 279 feet \( \times \) 0.3048 meters/foot = 85.0488 meters.

This unit conversion provides the necessary values in a standard metric measurement, essential for further calculations.

Rectangular Prism Volume

A rectangular prism is a three-dimensional shape with width, length, and depth as its dimensions. Calculating its volume tells you how much space it occupies in three-dimensional space.
The formula used is: \[ \text{Volume} = \text{Width} \times \text{Length} \times \text{Depth} \]This formula multiplies the area of the base (width \( \times \) length) by the height (depth) of the prism. For Lake Michigan, after converting its dimensions to meters:

• Width: 189901.32 meters
• Length: 494067.38 meters
• Depth: 85.0488 meters

To find the lake's volume, plug these values into the formula:\[ \text{Volume} = 189901.32 \text{ m} \times 494067.38 \text{ m} \times 85.0488 \text{ m} \]The calculated result gives an approximate volume for Lake Michigan.

Measurement Conversion

Measurement conversion is all about changing the numerical representation of units to help make computations easier and results more understandable. When working with different units such as miles, feet, and meters, conversion is necessary to put values on a common scale.
This process is often straightforward thanks to universal conversion factors. For example, converting from miles to meters uses the factor of 1609.34, while changing feet to meters uses 0.3048. Accurately applying these factors ensures that the subsequent calculations within a single measurement system make logical sense.
This practice extends beyond theoretical problems, helping solve real-world tasks, like estimating the volume of large bodies of water. Consistent measurements allow for precise volume calculations in fields like geography and hydrology.

Great Lakes Volume Estimation

Estimating the volume of the Great Lakes, such as Lake Michigan, involves applying principles of geometry and measurement. By using the rectangular prism model for volume calculation, we can achieve a reasonable approximation.
Despite having irregular shapes, these bodies of water are often simplified as prisms for initial calculations. Once dimensions are converted to a common unit like meters, you apply the volume formula:

• Width \( \times \) Length \( \times \) Depth

For Lake Michigan, the calculation might be:\[ \text{Volume} \approx 189901.32 \text{ m} \times 494067.38 \text{ m} \times 85.0488 \text{ m} \]This estimation gives an enormous volume, demonstrating the lake's vast water capacity. Such estimations are not only academic exercises but provide foundational data for environmental studies and resource management.