The volume of water used for crop irrigation is measured in acre-feet, where 1 acre-foot is the amount of water needed to cover 1 acre of land to a depth of 1 ft. (a) If there are 640 acres per square mile, how many cubic feet of water are in 1 acre-foot? (b) How many acre-feet are in Lake Erie (total volume = 116 mi^3)?

Verified step by step guidance

1. Step 1: To find the volume of water in 1 acre-foot, we need to convert the area of 1 acre to square feet. We know that 1 acre is equal to 43,560 square feet. Since 1 acre-foot covers this area to a depth of 1 foot, the volume in cubic feet is simply the area in square feet times the depth in feet. So, the volume of 1 acre-foot in cubic feet is 43,560 ft^2 * 1 ft = 43,560 ft^3.
2. Step 2: To find the number of acre-feet in Lake Erie, we first need to convert the volume of the lake from cubic miles to cubic feet. We know that 1 mile is equal to 5,280 feet, so 1 cubic mile is equal to 5,280 ft * 5,280 ft * 5,280 ft = 147,197,952,000 ft^3.
3. Step 3: Now, we can divide the total volume of Lake Erie in cubic feet by the volume of 1 acre-foot in cubic feet to find the total volume in acre-feet. So, the volume of Lake Erie in acre-feet is 116 mi^3 * 147,197,952,000 ft^3/mi^3 / 43,560 ft^3/acre-foot.
4. Step 4: Simplify the above expression to get the volume of Lake Erie in acre-feet.
5. Step 5: The result from step 4 is the volume of Lake Erie in acre-feet.

Key Concepts

Volume Conversion

Understanding volume conversion is essential for solving problems involving different units of measurement. In this case, converting acre-feet to cubic feet requires knowing the relationship between acres, feet, and cubic measurements. One acre-foot represents a volume that covers one acre of land to a depth of one foot, which can be calculated using the conversion factors for area and depth.

Area and Volume Relationships

The relationship between area and volume is crucial in this context, particularly when dealing with land and water measurements. An acre is a unit of area, and when multiplied by a depth (in feet), it yields a volume in cubic feet. This concept helps in visualizing how much water is needed to cover a specific area to a certain depth, which is fundamental for irrigation calculations.

Cubic Miles to Cubic Feet Conversion

To solve for the number of acre-feet in a volume given in cubic miles, one must understand how to convert cubic miles to cubic feet. Since 1 cubic mile equals 147,197,952,000 cubic feet, this conversion is necessary to relate the total volume of Lake Erie to acre-feet. This concept is vital for accurately translating large volume measurements into more manageable units for practical applications.