Problem 76 - 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 ft. An acre is 4840 yd. 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

(a) The cost of desalinated water per liter is \(\frac{\$1950}{(4840 \times (\frac{0.9144}{1})^2 \times \frac{0.3048}{1}) \times 1000}\) dollars per liter. (b) The cost per day for one household using only desalinated water is \(\frac{\$1950}{2 \times 365}\) dollars per day.

Step 1: Convert acre-foot to liters

We know that 1 acre-foot = 4840 yd^2 * 1 ft (depth), and we should convert it to liters. First, let's convert the volume from cubic yards to cubic meters: 1 yd = 0.9144 m, so 1 ft = 0.3048 m Thus, we can calculate the volume of an acre-foot in cubic meters: \[4840 \text{ yd}^2 \times 1 \text{ ft} \times (\frac{0.9144 \text{ m}}{1 \text{ yd}})^2 \times \frac{0.3048 \text{ m}}{1 \text{ ft}}\]

Step 2: Calculate the cost of desalinated water per liter

Now we know the volume of an acre-foot in cubic meters. We will convert it to liters (1 m^3 = 1000 L) and then calculate the cost of desalinated water per liter. \(Cost~per~liter = \frac{\$1950}{V}\), where V is the volume of an acre-foot in liters.

Step 3: Calculate the daily water consumption of a household

From the problem, 1 acre-foot of water is enough to supply two typical households for 1.00 year. We will find the amount of water consumed by one household daily. \(Daily~Water~Consumption = \frac{V}{2 \times 365}\)

Step 4: Calculate the cost for one household per day

Finally, we can calculate the cost of desalinated water for one household per day using cost per liter and daily water consumption: \(Cost~per~day = Cost~per~liter \times Daily~Water~Consumption\)

Key Concepts

Conversion of Units

When solving scientific or mathematical problems, particularly in fields like chemistry or physics, conversion of units is a vital skill. Part of understanding unit conversion involves translating one type of measurement into another in a practical and understandable way.

In our example, the challenge is to convert acreage measurement of water into liters. Starting from acre-feet, we first need to consider the area that water covers in land measurement units, converting it into a volume. An acre-foot is the volume of water that covers an acre of land to a depth of one foot.

• The area of an acre is given as 4840 square yards.
• The volume is then expressed in cubic yards by incorporating the foot depth.

After understanding the dimensions, proceed by converting each unit (yards and feet) into meters, as:

1 yard = 0.9144 meters and 1 foot = 0.3048 meters.

Applying these conversions, calculate the volume of an acre-foot in cubic meters. Finally, convert cubic meters to liters since 1 cubic meter equals 1000 liters. This results in the volume measurement needed to find out different cost factors.

Cost Calculation

After determining the volume in liters, the next step is calculating costs. This involves understanding how much you pay for a unit and extending this to larger quantities.

Our problem requires calculating the cost of desalinated water per liter from a price given for an entire acre-foot. The process requires the following steps:

• First, convert the total dollars for an acre-foot of water, here presented as $1950, into a price per liter.
• Once you have the volume from previous unit conversions, simply divide the total cost by this volume in liters to derive the cost per liter.

This conversion is crucial for day-to-day practical cost comparison and resource budgeting, ensuring accurate daily expenditures.

Household Water Consumption

Understanding household water consumption is essential for resource management and financial planning. Here, knowing how much a household typically uses can help calculate personal costs and conserve resources. An acre-foot of water, as stated in the problem, can supply two homes for a year.

To determine daily water usage, first find the total number of liters for one household in a year ([] for cubic meter conversion details if needed). Then simply divide by the number of days in a year (365) to find daily consumption. Break it down further by considering half of the acre-foot for one house:

• Determine the annual volume divided by 2, representing two households sharing one acre-foot.
• Daily usage then comes from further dividing by 365, indicating the consumption per household each day.

This value is crucial during cost calculation and sustainable planning, ensuring you know how much you're consuming daily.