Problem 70 - Chemistry: The Central Science

Question

In the United States, water used for irrigation is measured in acre-feet. An acre-foot of water covers an acre to a depth of exactly \(1 \mathrm{ft}\). An acre is \(4840 \mathrm{yd}^{2}\). An acre-foot is enough water to supply two typical households for \(1.00 \mathrm{yr}\). (a) If desalinated water costs \(\$ 1950\) per acre-foot, how much does desalinated water cost per liter? (b) How much would it cost one household per day if it were the only source of water?

Step-by-Step Solution

Answer

The cost of desalinated water per liter is approximately \(0.158\) cents/L, and the daily cost for one household if it were the only source of water would be approximately $2.67.

Step 1: Find the volume of one acre-foot

To find the volume of one acre-foot of water, we need to convert the area of an acre and the depth of 1 ft into the same unit of measurement. Since we are asked to find the cost per liter, it's best to convert everything in metric units. First, we convert the area of an acre into square meters and the depth into meters: 1 acre = 4840 yd² 1 yd = 0.9144 m (conversion factor) Area = 4840 yd² × (0.9144 m/yd)² = \(\approx\) 4046.86 m² 1 ft = 0.3048 m (conversion factor) Depth = 1 ft × (0.3048 m/ft) = 0.3048 m Now, we can find the volume of one acre-foot by multiplying the area with the depth: Volume = Area × Depth = 4046.86 m² × 0.3048 m = \(\approx\) 1233.48 m³ To convert cubic meters into liters, we use the conversion factor 1 m³ = 1000 L: Volume = 1233.48 m³ × 1000 L/m³ = \(\approx\) 1233480 L

Step 2: Calculate the cost per liter

We are given that desalinated water costs $1950 per acre-foot. Since we have now found the volume of one acre-foot, we can find the cost per liter. Cost per liter = (Cost per acre-foot) / (Volume in liters) = (1950 USD) / (1233480 L) \(\approx\) 0.00158 USD/L As most currency calculations are made to the nearest cent, we may want to express this in cents per liter: 0.00158 USD/L × 100 cents/USD \(\approx\) 0.158 cents/L

Step 3: Calculate the daily cost for one household

We are given that an acre-foot is enough water to supply two typical households for one year, so we need to find out how much water one household requires per day. Days per year = 365 days Water per household per year = Volume / 2 households = 1233480 L / 2 \(\approx\) 616740 L/household Water per household per day = Water per household per year / Days per year = 616740 L / 365 days \(\approx\) 1688.6 L/day Now we can calculate the cost for one household per day using the cost per liter: Cost per day = Water per household per day × Cost per liter = 1688.6 L/day × 0.00158 USD/L \(\approx\) 2.67 USD/day So, the daily cost for one household if desalinated water were the only source of water would be approximately $2.67.

Key Concepts

Acre-Foot

An acre-foot is a unit used to measure large volumes of water, especially in the United States for purposes like irrigation. It represents the volume of water that would cover one acre of land to a depth of one foot. To understand this better:

• An acre is an area measurement. It is equivalent to 4,840 square yards or 43,560 square feet.
• A foot is a measurement of length. In this context, it refers to the depth the water would reach.

The combination of these measurements gives us the total volume covered, which is crucial for calculating water usage in agriculture and understanding the capacity needed to supply households or farms.

Desalination Cost

Desalination is the process of removing salt and impurities from seawater to make it suitable for drinking and irrigation. The cost of this process can be significant and is often expressed per unit of water, such as an acre-foot in this scenario. With the given information, the cost of desalinated water is $1950 per acre-foot. By converting this into cost per liter, we get a clearer sense of the expense associated with desalination in smaller, more practical terms, which can help in comparing it with other water sources or budgeting for household or agricultural use.

Volume Conversion

Volume conversion refers to changing a measurement from one unit to another. In our problem, we move between units commonly used in the United States and the metric system. To determine the volume of an acre-foot in a metric unit (cubic meters and then liters):

• Convert the area of an acre to square meters using the conversion 1 yard = 0.9144 meters.
• Calculate the volume by multiplying this area by the depth in meters (1 foot = 0.3048 meters).
• Finally, convert cubic meters to liters knowing that 1 cubic meter equals 1,000 liters.

This allows us to express volumes in liters, which are more commonly used in many contexts, improving comprehension and usability of data.

Metric Conversion

Metric conversion is the process of changing from one set of measurement units to another, particularly to metric units, which are widely used in science and many international standards. In the context of this exercise:

• Length and area are converted from yards and feet into meters and square meters.
• This facilitates clearer comparison and ensures that volumes like acre-feet can be universally understood through conversions to liters.
• The conversion factors, such as 1 yard = 0.9144 meters and 1 foot = 0.3048 meters, are vital to ensuring accuracy in these transformations.

Understanding these conversions is essential for anyone studying or working internationally, ensuring consistency and understanding across different measurement systems.