a farmer has 2400 ft. of fencing and wants to fence off a rectangular field that borders a straight river…

a farmer has 2400 ft. of fencing and wants to fence off a rectangular field that borders a straight river. he needs no fencing along the river. what are the dimensions of the field that has the largest area?

a farmer has 2400 ft. of fencing and wants to fence off a rectangular field that borders a straight river. he needs no fencing along the river. what are the dimensions of the field that has the largest area?

Answer

Explanation:

Step1: Define variables

Let the length parallel to the river be (x) and the width perpendicular to the river be (y). The total fencing is (x + 2y=2400), so (x = 2400 - 2y).

Step2: Express the area

The area (A=xy). Substitute (x = 2400 - 2y) into the area formula: (A=(2400 - 2y)y=2400y-2y^{2}).

Step3: Find the derivative of the area function

Differentiate (A(y)) with respect to (y). (A^\prime(y)=\frac{d}{dy}(2400y - 2y^{2})=2400-4y).

Step4: Set the derivative equal to zero

Set (A^\prime(y) = 0), then (2400-4y = 0). Solve for (y): (4y=2400), (y = 600).

Step5: Find (x)

Substitute (y = 600) into (x = 2400 - 2y). (x=2400-2\times600=1200).

Answer:

The dimensions are (x = 1200) ft (parallel to the river) and (y = 600) ft (perpendicular to the river).