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 acrefoot is enough water to supply two typical households for \(1.00\) yr. (a) If desalinated water costs \(\$ 1950\) per acrefoot, 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?

Short answer

(a) The cost per liter of desalinated water is \(\dfrac{\$ 1950}{\big(\dfrac{1 \cdot 4840 \mathrm{yd}^{2}\cdot 1 \mathrm{ft}}{27}\cdot 764.6 \mathrm{L/ yd}^{3}\big)} \approx\$ 0.0016\). (b) The cost per household per day for using desalinated water is \(\dfrac{\dfrac{\$ 1950}{2}}{365} \approx\$ 2.67 \).

Step-by-step solution

Convert acre-feet to cubic feet

To convert the given acre-feet of water to cubic feet, remember that 1 acre-foot covers an acre to a depth of 1 foot, so the volume in cubic feet is: \(V_\mathrm{ft} = 1 \cdot A_\mathrm{acre}\cdot 1 \mathrm{ft}\) where \(A_\mathrm{acre}\) is the area given in square yards.

Convert acre to square yards

We are given that 1 acre is 4840 square yards. Substitute this value in the formula above: \(V_\mathrm{ft} = 1 \cdot 4840 \mathrm{yd}^{2}\cdot 1 \mathrm{ft}\)

Convert cubic feet to cubic yards

We need to convert the volume in cubic feet to cubic yards to match the unit of acre. There are 27 cubic feet in 1 cubic yard, so we can now convert the volume: \(V_\mathrm{yd} = \dfrac{V_\mathrm{ft}}{27}\) Substitute the volume in cubic feet from step 2: \(V_\mathrm{yd} = \dfrac{1 \cdot 4840 \mathrm{yd}^{2}\cdot 1 \mathrm{ft}}{27}\)

Convert cubic yards to liters

Now let's convert the volume in cubic yards to liters. There are approximately 764.6 liters in a cubic yard, so multiply the volume by this conversion factor: \(V_\mathrm{L} = V_\mathrm{yd} \cdot 764.6 \mathrm{L/ yd}^{3}\) Substitute the volume in cubic yards from step 3: \(V_\mathrm{L} = \dfrac{1 \cdot 4840 \mathrm{yd}^{2}\cdot 1 \mathrm{ft}}{27} \cdot 764.6 \mathrm{L/ yd}^{3}\)

Calculate the cost per liter

We are given that the cost of desalinated water per acre-foot is $1950. Now divide that cost by the total liters found in step 4: Cost per liter = \(\dfrac{\$ 1950}{V_\mathrm{L}}\) Substitute the volume in liters from step 4: Cost per liter = \(\dfrac{\$ 1950}{\big(\dfrac{1 \cdot 4840 \mathrm{yd}^{2}\cdot 1 \mathrm{ft}}{27}\cdot 764.6 \mathrm{L/ yd}^{3}\big)}\) Calculate the cost per liter.

Calculate the cost per household per year

We are given that 1 acre-foot of water can supply two households for a year. So, divide the cost per acre-foot by 2 to find the cost per household per year: Cost per household per year = \(\dfrac{\$ 1950}{2}\) Calculate the cost per household per year.

Calculate the cost per household per day

Now divide the cost per household per year calculated in step 6 by the number of days per year (assume 365 days for simplicity): Cost per household per day = \(\dfrac{\text{Cost per household per year}}{365}\) Substitute the cost per household per year from step 6 and calculate the result.