Question

Find the greatest possible enclosed area of a rectangular corral given 400 feet of fencing

Answer 1

The correct answer is 1000 Square feet

Answer 2

Maximum area = 10000 square units.

Explanation:

We are given the following information:

Rectangular perimeter of coral = 400 units.

Let length of the coral be x. Then,

Perimeter = 400 = 2(Length +Breadth)

`400 = 2(x + Breadth)\\Breadth = 200 - x`

Thus, the area of rectangle is given by,

`Area = Length* Breadth = x* (200-x) = 200x - x^2`

Thus, we have to maximize the function:

`f(x) = 200x - x^2`

We will use double derivative test.

First we differentiate with respect to x.

`\displaystyle(d(f(x)))/(dx) = \displaystyle(d(200x - x^2))/(dx) = 200 - 2x`

Equating this to zero to obtain critical points,

`200 - 2x = 0\\200 = 2x\\x = 100`

Now, again differentiating with respect to x.

`\displaystyle(d^2(f(x)))/(dx^2) = -2 < 0`

Thus, by double derivative test, local maxima occurs for this function at x = 100

So, Length = x = 100 units

Breadth = 200 - x = 100 units

Maximum area = 10000 square units.