a rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a…
a rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single - strand electric fence. with 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? the maximum area of the rectangular plot is
Answer
Explanation:
Step1: Define variables
Let the length of the side parallel to the river be (x) meters and the length of the other two sides be (y) meters. The total length of the fence is (x + 2y=800), so (x = 800 - 2y). The area (A=xy).
Step2: Substitute (x) into the area formula
Substitute (x = 800 - 2y) into (A = xy), we get (A=(800 - 2y)y=800y-2y^{2}).
Step3: Find the derivative of (A) with respect to (y)
Differentiate (A(y)=800y - 2y^{2}) using the power rule ((x^{n})^\prime=nx^{n - 1}). (A^\prime(y)=800-4y).
Step4: Set the derivative equal to zero and solve for (y)
Set (A^\prime(y)=0), so (800-4y = 0). Then (4y=800), and (y = 200).
Step5: Find (x)
Substitute (y = 200) into (x=800 - 2y), we get (x=800-2\times200 = 400).
Step6: Calculate the maximum area
Substitute (x = 400) and (y = 200) into (A=xy), (A=400\times200=80000) (m^{2}).
Answer:
(80000) (m^{2})