eddiecasas247
contestada

Find the dimensions of the rectangular corral producing the greatest enclosed area given 320 feet of fencing. (Assume that the length is greater than or equal to the width.)

Respuesta :

calculista

Answer:

The dimensions of the rectangular corral producing the greatest enclosed area  is a square of 80 ft x 80 ft

Step-by-step explanation:

Let

x -----> the length of the rectangular corral in feet

y -----> the width of the rectangular corral in feet

we know that

The area of the rectangular corral is equal to

[tex]A=xy[/tex] -----> equation A

The perimeter of the rectangular corral is equal to

[tex]P=2(x+y)[/tex]

[tex]P=320\ ft[/tex]

so

[tex]320=2(x+y)[/tex]

Simplify

[tex]160=(x+y)[/tex]

[tex]y=160-x[/tex] -----> equation B

substitute equation B in equation A

[tex]A=x(160-x)[/tex]

[tex]A=-x^2+160x[/tex]

This is a vertical parabola open downward

The vertex is a maximum

The x-coordinate of the vertex represent the length for the maximum area

The y-coordinate of the vertex represent the maximum area

Convert the quadratic equation in vertex form

[tex]A=-x^2+160x[/tex]

Factor -1

[tex]A=-(x^2-160x)[/tex]

Complete the square

[tex]A=-(x^2-160x+80^2)+80^2[/tex]

[tex]A=-(x^2-160x+80^2)+6,400[/tex]

Rewrite as perfect squares

[tex]A=-(x-80)^2+6,400[/tex]

The vertex is the point (80,6,400)

so

[tex]x=80\ ft[/tex]

The maximum area is 6,400 ft^2

Find the value of y

[tex]y=160-x[/tex]  ----> [tex]y=160-80=80\ ft[/tex]

therefore

The dimensions of the rectangular corral producing the greatest enclosed area  is a square of 80 ft x 80 ft

The dimension of the rectangular corral is 80 by 80 feet. when the length is equal to the width.

What is a rectangle?

It is a polygon that has four sides. The sum of the internal angle is 360 degrees. In a rectangle, opposite sides are parallel and equal and each angle is 90 degrees. And its diagonals are also equal and intersect at mid-point.

The perimeter of the rectangle is 320 feet.

Let L be the length and W be the width of the rectangle.

Assume that the length is greater than or equal to the width. Then

[tex]\rm L \geq W[/tex]

Then we have, the perimeter of the rectangle is given as

[tex]\rm Perimeter = 2( Length + Width)\\[/tex]

Then

[tex]\rm 2(L+W) = 320\\\\L \ \ + \ \ W = 160[/tex]

Let L = W, then

[tex]\rm W = 80[/tex]

Thus, the dimension of the rectangular corral is 80 by 80 feet.

More about the rectangle link is given below.

https://brainly.com/question/10046743

Otras preguntas