A farmer is fencing a rectangular area for cattle and uses a straight portion of a river as one side of the rectangle, as illustrated in the figure. Note that there is no fence along the river. If the farmer has 1000 feet of fence, find the dimensions for the rectangular area that gives the maximum area for the cattle.
Guide On Rating System
Vote
The farmer has 1000 feet of fence to use for the three sides of the rectangle. Let's denote the length of the rectangle parallel to the river as x and the length of the rectangle perpendicular to the river as y.
Since the farmer has 1000 feet of fence, the total length of the fence used for the rectangle is x + 2y = 1000.
We can solve this equation for y to get y = (1000 - x) / 2.
The area A of the rectangle is given by A = x * y. Substituting y from the equation above, we get A = x * (1000 - x) / 2 = 500x - x^2/2.
To find the maximum area, we need to find the maximum of this function. The maximum of a function occurs where its derivative is zero.
The derivative of A with respect to x is A' = 500 - x. Setting this equal to zero gives x = 500.
Substituting x = 500 back into the equation for y gives y = (1000 - 500) / 2 = 250.
So, the dimensions that give the maximum area are x = 500 feet and y = 250 feet.
Got a question with my answer?
Cheapmath.com is completely free!
Asking question is free.
Getting answers is free.
+99