Problem 76 - Chemistry: The Central Science in SI Units, Global Edition

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 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 per liter of desalinated water is approximately $0.00158/liter, and the daily cost per household for using desalinated water as the only source is about $2.67 per day.

Step 1: Convert acre to square meters

First, we need to convert an acre to square meters. We know that an acre is 4840 square yards. To convert it to square meters, we use the fact that 1 yard = 0.9144 meters. 1 acre = 4840 sq.yards * (0.9144 m/yd)^2 1 acre = 4840 * (0.9144)^2 sq.meters 1 acre ≈ 4046.86 sq.meters

Step 2: Calculate volume in acre-feet

Now we will find the volume of water in an acre-foot. Since the depth is 1 foot and 1 foot is equal to 0.3048 meters, we have: Volume = Area * Depth Volume = 4046.86 sq.meters * 0.3048 meters Volume ≈ 1233.49 cubic meters

Step 3: Convert cubic meters to liters

We need to convert the volume to liters. We know that 1 cubic meter = 1000 liters. Volume in liters = 1233.49 cubic meters * 1000 liters/cubic meter Volume in liters ≈ 1,233,490 liters

Step 4: Calculate the cost per liter

Now we can find the cost of desalinated water per liter. The cost per acre-foot is given as $1950. Cost per liter = \(\frac{1950}{1233490}\) dollars/liter Cost per liter ≈ \(0.00158\) dollars/liter

Step 5: Calculate the daily cost per household

We know that one acre-foot of water can supply two households for a year. So, to find the cost per household per day, we will divide the cost per acre-foot by 2 (number of households) and then by 365 (days in a year). Cost per household per day = \(\frac{1950}{2 \times 365}\) dollars/day Cost per household per day ≈ $2.67/day The cost per liter of desalinated water is approximately \(0.00158/liter, and the daily cost per household for using desalinated water as the only source is about \)2.67 per day.

Key Concepts

Acre-Foot

When discussing water usage, especially in agricultural settings in the United States, the term "acre-foot" frequently comes up. But what exactly does it mean? An acre-foot represents the volume of water necessary to cover one acre of land to the depth of one foot. This measurement is crucial because it provides a standard way to quantify large volumes of water.

• 1 acre-foot covers 1 acre of land to a depth of 1 foot.
• It is approximately equal to 1233.49 cubic meters.
• This measurement can supply water to two average households for a year.

Understanding the acre-foot helps in managing water resources effectively, making it easier to compare and manage water consumption across different regions and sectors.

Volume Calculation

Calculating the volume of water involved in an acre-foot requires some basic conversion skills. We need to know the area and the depth to calculate volume. Here's how we do it:

• An acre equals 4840 square yards.
• Converting square yards to square meters: 1 yard = 0.9144 meters.
• Therefore, 1 acre ≈ 4046.86 square meters.
• The depth of 1 acre-foot is 1 foot, which equals 0.3048 meters.

Now, we calculate the volume using the formula: Volume = Area × Depth. Plug in the values for an acre-foot:

• Volume = 4046.86 sq.meters × 0.3048 meters ≈ 1233.49 cubic meters.
• Convert cubic meters to liters (1 cubic meter = 1000 liters):
  • Volume = 1233.49 cubic meters × 1000 liters/cubic meter = 1,233,490 liters.

By understanding and performing these calculations, we can accurately translate large measurements into more usable units, like liters, which are practical for everyday use.

Cost Analysis

When it comes to water usage, understanding the cost associated with it is equally as important as understanding the volume. In our exercise, desalinated water costs \(1950 per acre-foot. To determine the cost per liter, we take the total cost and divide it by the volume in liters.

• Cost per liter = \(\frac{1950}{1233490}\) dollars/liter.
• Cost per liter ≈ \)0.00158.

Next, understanding the daily expense for a household can be crucial for budgeting water resources:

• 1 acre-foot serves two households for a year, which means 0.5 acre-foot per household per year.
• Daily cost per household = \(\frac{1950}{2 \times 365}\) dollars/day ≈ $2.67.

This detailed cost analysis helps households and planners make informed decisions on water budgeting, especially in areas reliant on desalinated water. By calculating both the cost per liter and household expense, we gain insights into daily costs which are practical for planning and conserving water resources.