Question

A farmer is building a fence to enclose a rectangular area against an existing wall, shown in the figure below.

Answer 1

The most appropriate choice for maxima and minima of a function will be given by

Rectangle of length 72 feet and breadth 36 feet has largest area

What is maxima and minima?

Maxima of function f(x) is the maximum value of the function and minima of function f(x) is the minimum value of the function.

Here,

Let the length be x feet and breadth be y feet

The farmer has 144 feet of fencing

Three of the sides will require fencing and the fourth wall already exists.

So,

x + y + y =1 44

x + 2y = 144

Area of rectangle(A) = xy $ft^2$

                               = (144 - 2y)y

                               = $144y - 2y^2$

$\frac{dA}{dy} = \frac{d}{dy}(144y - 2y^2)$

     = $144 - 2\times 2y^{2-1}\\144 - 4y$

For largest area,

$\frac{dA}{dy} = 0$

$144 - 4y = 0 \\4y = 144\\y = \frac{144}{4}\\y = 36$

$\frac{d^2A}{dy^2} = \frac{d}{dy}(144 - 4y)\\=0-4\\=-4 < 0$

Hence area is maximum

For largest area, y = 36 feet

$x = 144 - 2\times 36\\x = 144-72$

$x = 72$ feet

So length of rectangle is 72 feet, breadth of rectangle is 36 feet

To learn more about maxima and minima of a function, refer to the link:

brainly.com/question/14378712

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