Question

jenna's rectangular garden borders a wall. she buys 80 ft of fencing. what are the dimensions of the garden that will maximize its area?

Answer 1

Answer:

The  dimensions are x =20 and y=20 of the garden that will maximize its area is 400

Step-by-step explanation:

Step 1:-

let 'x' be the length  and the 'y' be the width of the rectangle

given Jenna's buys 80ft of fencing of rectangle so the perimeter of the rectangle is    2(x +y) = 80

                         x + y =40

                               y = 40 -x

now the area of the rectangle A = length X width

                                                  A = x y

substitute 'y' value in above A = x (40 - x)

                                              A = 40 x - x^2 .....(1)

Step :2

now differentiating equation (1) with respective to 'x'

                                      $\frac{dA}{dx} = 40 -2x$     ........(2)

Find the dimensions

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

40 - 2x =0

40 = 2x

x = 20

and y = 40 - x = 40 -20 =20

The dimensions are x =20 and y=20

length = 20 and breadth = 20

Step 3:-

we have to find maximum area

Again differentiating equation (2) with respective to 'x' we get

$\frac{d^2A}{dx^2} = -2$

Now the maximum area A =  x y at x =20 and y=20

                                        A = 20 X 20 = 400

                                         

Conclusion:-

The  dimensions are x =20 and y=20 of the garden that will maximize its area is 400

verification:-

The perimeter = 2(x +y) =80

                           2(20 +20) =80

                              2(40) =80

                              80 =80