A lawn sprinkler covers a rectangular pattern of 20ft x 30ft.
1)If the flow rate of the sprinkler is 2.5 gpm, derive a formula for calculating the time (hours) to apply D inches of water over one 20ft x 30ft area.
2)What are the units on the constant?
20ft*30ft*1in = 600ft^2-in
1 gal = 231 in^3
2.5gal = 577.5 in^3
1 ft^2-in = 144 in^3, so
2.5gal/min = 4.04 ft^2-in/min
or, .24935 min/ft^2 in
that's 149.61 min/600 ft^2-in
or, 2.4935 hr/600 ft^2-in
so, you will need roughly 2.5D hours to cover the area with D inches of water
since your formula is time(hr) = k*D(in), k must be in hr/in
To derive a formula for calculating the time to apply a certain depth of water, we first need to determine the volume of water needed to cover the given area.
1) Volume of water needed:
The area of the rectangular pattern is 20ft x 30ft = 600 square feet.
To convert this into cubic feet, we need to multiply the area by the depth (D) in feet.
Volume = Area x Depth
Volume = 600ft² x Dft = 600D ft³
Since we are given the flow rate in gallons per minute (gpm), we need to convert the volume from cubic feet to gallons. 1 cubic foot is approximately equal to 7.48 gallons.
Volume = 600D ft³ x 7.48 gallons/ft³ = 4,488D gallons
The next step is to calculate the time required to apply this volume of water based on the given flow rate.
2.5 gallons per minute means that the sprinkler can distribute 2.5 gallons of water in 1 minute.
Time (in minutes) = Volume / Flow rate
Time (in minutes) = 4,488D gallons / 2.5 gallons per minute = 1795.2D minutes
However, the question asks for the time in hours, so we need to convert the minutes to hours by dividing by 60.
Time (in hours) = 1795.2D minutes / 60 minutes per hour = 29.92D hours
Therefore, the formula for calculating the time (in hours) to apply D inches of water over one 20ft x 30ft area is:
Time (in hours) = 29.92D
2) The units on the constant, 29.92, are "hours per inch" or "hours/inch". This means that for each inch of water, it takes approximately 29.92 hours to apply that amount of water over the given area.