Question

Alexandra is installing edge material around her yard. She has 400 ft of edge material to surround three sides of her rectangular yard. The fourth side will be against her deck and does not need edging. What is the maximum area that can be enclosed by the edging? Enter your answer in the box.

Answer 1

Answer:

The maximum area is equal to $20,000\ ft^{2}$

Step-by-step explanation:

Let

x ----> the length of the rectangular yard

y ----> the width of the rectangular yard

we know that

The perimeter is equal to

$400=x+2y$ --> remember that the fourth side will be against her deck

isolate the variable y

$y=200-0.5x$ -----> equation A

The area of the rectangular yard is equal to

$A=xy$ ----> equation B

substitute equation A in equation B

$A=x(200-0.5x)\\ \\A=200x-0.5x^{2}$

The quadratic function is a vertical parabola open downward

The vertex is a maximum

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

The y-coordinate of the vertex represent the maximum area of the rectangular yard

Using a graphing tool

The vertex is the point (200,20,000)

see the attached figure

therefore

The length of the rectangular yard is 200 ft

The width of the rectangular yard is $y=200-0.5(200)=100\ ft$

The maximum area is equal to $20,000\ ft^{2}$