Problem 47 - Physics for Scientists and Engineers with Modern Physics

Question

An average family of four uses roughly \(1200 \mathrm{~L}\) (about 300 gallons) of water per day \(\left(1 \mathrm{~L}=1000 \mathrm{~cm}^{3}\right)\). How much depth would a lake lose per year if it uniformly covered an area of \(50 \mathrm{~km}^{2}\) and supplied a local town with a population of 40,000 people? Consider only population uses, and neglect evaporation and so on.

Step-by-Step Solution

Answer

The lake would lose 8.76 cm of depth per year.

Step 1: Calculate Daily Water Usage for the Town

First, determine the total daily water usage for the entire town. If an average family of four uses 1200 L of water per day, then one person uses \( \frac{1200}{4} = 300 \text{ L} \) per day. With a town population of 40,000 people, the total daily water usage is \( 40,000 \times 300 = 12,000,000 \text{ L} \).

Step 2: Calculate Annual Water Usage

Next, calculate the total water usage for the entire year. Multiply the daily usage by the number of days in a year: \( 12,000,000 \times 365 = 4,380,000,000 \text{ L} \).

Step 3: Convert Liters to Cubic Meters

Convert the annual water usage from liters to cubic meters, knowing that \(1 \text{ L} = 0.001 \text{ m}^3 \). So, \( 4,380,000,000 \text{ L} = 4,380,000 \text{ m}^3 \).

Step 4: Determine the Loss in Lake Depth

Finally, calculate how much depth the lake loses. The lake covers an area of 50 km², which is \(50,000,000 \text{ m}^2 \). Divide the annual water usage by the area to find the depth loss: \( \frac{4,380,000 \text{ m}^3}{50,000,000 \text{ m}^2} = 0.0876 \text{ m} \), or 8.76 cm.

Key Concepts

Water Usage Calculation

Understanding water usage is essential when analyzing how resources are consumed in our daily lives. In the given problem, we start by calculating water usage for a typical family and then extend it to a larger population. First, consider that an average family of four uses 1200 liters per day. By this calculation, we can determine how much one person uses: simply divide 1200 liters by 4, which gives us 300 liters per person.

Once you have an individual's daily water use, you can scale it up to a larger group or town. Multiply the daily water usage per person by the total population, here exemplified as 40,000 people. So, the town uses 12 million liters each day.

These calculations form the basis for understanding broader water management issues and allow accurate planning for supply requirements. Ultimately, knowing the total daily consumption is the first step toward figuring out longer-term consumption, such as annually or seasonally.

Unit Conversion

Unit conversion is a vital skill in physics to ensure all quantities are in the correct units for further analysis. This problem involves converting between different units to solve for the depth loss in a lake. Let's see how it's done:

Initially, daily water usage is given in liters. To analyze the annual usage, multiply the daily amount by 365 days. The result is a vast figure, making it suitable to convert into cubic meters for convenience.

• 1 liter is defined as 0.001 cubic meters. Therefore, converting liters to cubic meters involves multiplying by this factor.
• In our example, 4,380,000,000 liters becomes 4,380,000 cubic meters.

Correct unit conversion ensures you can proceed with calculations accurately, ensuring figures are manageable and interpretable.

Area and Volume Calculations

Calculating area and volume is fundamental, especially when analyzing environmental factors such as resource consumption. In this problem, we calculate the water volume the lake needs to supply a town and determine the effect on the lake's depth.

Volume calculations in this context involve understanding the relationship between area and depth. The lake covers 50 square kilometers, which must be converted into square meters for compatibility with cubic meters. Knowing that 1 km² equals 1,000,000 m², the lake has an area of 50,000,000 m².

To find out how much depth is lost annually, we divide the volume of water used by the area of the lake. In this instance, it involves dividing 4,380,000 cubic meters by 50,000,000 square meters. The result shows that the lake loses approximately 8.76 cm per year.

Understanding these calculations helps in assessing the sustainability of water sources and planning for community needs efficiently.