Question

Marquise has 200 meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width w w (in meters) is modeled by: A ( w ) = − w 2 + 1 0 0 w A(w)=−w 2 +100w What side width will produce the maximum garden area? meters

Answer 1

Answer:

50 m

Step-by-step explanation:

We are given that

Width of garden =w

Perimeter of rectangular garden=200 m

We know that

Perimeter of rectangle=$2(l+ b)$

$200=2(L+w)$

$l+w=\frac{200}{2}=100$

$L=100-w$

Area of garden $A(w)=-w^2+100w$

We have to find the side width that will produce the maximum garden area.

Differentiate w.r.t w

$\frac{d(A)}{dw}=-2w+100$

Substitute $\frac{dA}{dw}=0$

$-2w+100=0$

$2w=100$

$w=\frac{100}{2}=50$

Again differentiate w.r.t w

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

Hence, the garden area is maximum at w=50 m

Therefore, width of rectangular garden=50 m

Answer 2

To find the optimized value you need to find where the vertex(tip) of the parabola

use
$- \frac{b}{2a}$
to find the optimized x value

y= -w^2 +100w + 0

$- \frac{100}{ 2 \times - 1}$
x = 50
answer: 50