Respuesta :

LammettHash
Let [tex]x[/tex] and [tex]y[/tex] be the sides of the rectangle. The perimeter is given to be 500m, so we are maximizing the area function [tex]A(x,y)=xy[/tex] subject to the constraint [tex]2x+2y=500[/tex].

From the constraint, we find

[tex]2x+2y=500\implies x+y=250\implies y=250-x[/tex]

so we can write the area function independently of [tex]y[/tex]:

[tex]A(x,y)=\hat A(x)=x(250-x)=250x-x^2[/tex]

Differentiating and setting equal to zero, we find one critical point:

[tex]\dfrac{\mathrm d\hat A(x)}{\mathrm dx}=250-2x=0\implies x=125[/tex]

which means [tex]y=250-125=125[/tex], so in fact the largest area is achieved with a square fence that surrounds an area of [tex]A(125,125)=125^2=15625\text{ m}^2[/tex].
raphealnwobi

The maximum area that can be enclosed is 15625 feet²

A rectangle is a quadrilateral in which opposite sides are equal and parallel to each other.

Let x represent the length and y represent the width. Hence:

perimeter = 2(length + width)

500 = 2(x + y)

250 = x + y

y = 250 - x

The area of a rectangle (A) = length * width

A = x * (250 - x)

A = 250x - x²

The maximum area is at dA/dx = 0, hence:

dA/dx = 250 - 2x

250 - 2x = 0

2x = 250

x = 125 feet

y = 250 - x = 250 - 125 = 125 feet

The maximum area = x * y = 125 * 125 = 15625 feet²

The maximum area that can be enclosed is 15625 feet²

Find out more at: https://brainly.com/question/11906003

Otras preguntas