Enclosing an Area

Question

Enclosing an Area 

Patricia wishes to have a rectangular garden in her backyard. She has 60 ft of fencing with which to enclose her garden.

A rectangle is shown. The width is labeled x and the length is labeled y.

Letting x denote the width of the garden, find a function f in the variable x giving the area of the garden.

f(x) = 

 

 (Expand your answer completely.)

What is its domain? (Enter your answer using interval notation.)

Answer 1

Hi Mark!

We have 60 ft of fencing. We know the

perimeter of the garden is 2x + 2y, so

2x + 2y = 60

We want a function in terms of x, so

solve for y.

2x + 2y = 60

-2x -2x

-------------------

2y = -2x + 60

--- ---- -----

2 2 2

y = -x + 30

The area of the rectangular

garden is A = x * y, or in

terms of x only

A = x * (-x + 30)

or A = -x² + 30x

To find the domain, find zeros of the

function. Best do do this in factored form..

x (-x+30) = 0

x = 0 or x = 30

At x = 0, you won't have a width

At x = 30, you won't have a length

Your domain will be between these

zeros. In interval notation: (0,30)

If you wanted to maximize the area

given the amount of fencing you have,

find the x-coordinate of the vertex:

x = -b/(2a)

A = -1x² + 30x

a b

x = -(30)/(2*-1)

x = 15ft

The length is then

y = -x + 30

y = -15 + 30 = 15ft

This makes sense that the length

and the width need to be the same

to maximize the area of a rectangle

(a square).