Question

Tyler has 120 linear feet of fencing to put around his garden. If he uses the fence to border a rectangular garden, what is the maximum area his garden can be? Answer should have units written out such as yards, square centimeters or cubic inches.

Answer 1

Answer:

The maximum area is $900\ ft^{2}$

Step-by-step explanation:

Let

x----> the length of rectangle

y---> the width of rectangle

we know that

The perimeter of rectangle is equal to

$P=2(x+y)$

we have

$P=120\ ft$

so

$120=2(x+y)$

$60=(x+y)$

$y=60-x$------> equation A

Remember that

The area of rectangle is equal to

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

substitute equation A in equation B

$A=x(60-x)$

$A=-x^{2} +60x$

This is a vertical parabola open downward

The vertex is a maximum

The y-coordinate of the vertex of the graph is the maximum area of the garden and the x-coordinate is the length for the maximum area

using a graphing tool

The vertex is the point $(30,900)$

see the attached figure

Find the value of y

$y=60-x$ ----->  $y=60-30=30\ ft$

The dimensions of the rectangular garden is $30\ ft$ by $30\ ft$

For a maximum area the garden is a square

The maximum area is $900\ ft^{2}$