contestada

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

Respuesta :

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

use
[tex] - \frac{b}{2a} [/tex]
to find the optimized x value

y= -w^2 +100w + 0

[tex] - \frac{100}{ 2 \times - 1} [/tex]
x = 50
answer: 50

lublana

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=[tex]2(l+ b)[/tex]

[tex]200=2(L+w)[/tex]

[tex]l+w=\frac{200}{2}=100[/tex]

[tex]L=100-w[/tex]

Area of garden [tex]A(w)=-w^2+100w[/tex]

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

Differentiate w.r.t w

[tex]\frac{d(A)}{dw}=-2w+100[/tex]

Substitute [tex]\frac{dA}{dw}=0[/tex]

[tex]-2w+100=0[/tex]

[tex]2w=100[/tex]

[tex]w=\frac{100}{2}=50[/tex]

Again differentiate w.r.t w

[tex]\frac{d^2A}{dw^2}=-2 <0[/tex]

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

Therefore, width of rectangular garden=50 m

Otras preguntas