Question

A rancher needs to enclose two adjacent rectangular​ corrals, one for cattle and one for sheep. if the river forms one side of the corrals and 330330 yd of fencing is​ available, find the largest total area that can be enclosed.

Answer 1

Given that the river encloses one of the side of the corals, then only three sides of the corals needs to be fenced and the demacation of the two sections.

The sides to be fenced is three equal sides plus one other side. Let the length of the coral be x, then the perimeter of the sides to be fenced is given by 3x + the other side = 330

Thus, the sides sides of the rectangle formed by the two adjacent corals are x and 330 - 3x.

The area of a rectangle is given by length x width.

Thus, the area of the corals to be enclosed is given by


`Area=x(330-3x)=330x-3x^2`

For the area to be maximum, the differentiation of the area with rexpect to x will equal 0.

Thus,

`(dA)/(dx) =330-6x=0 \\ \\ \Rightarrow6x=330 \\ \\ \Rightarrow x=55`

Therefore, the largest total area that can be enclosed is given by


`Area=330x-3x^2 \\ \\ =330(55)-3(55)^2 \\ \\ =18,150-9,075 \\ \\ =9,075\ yd^2`