Respuesta :
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'
[tex]\frac{dA}{dx} = 40 -2x[/tex] ........(2)
Find the dimensions
[tex]\frac{dA}{dx} = 0[/tex]
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
[tex]\frac{d^2A}{dx^2} = -2 <0[/tex]
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
