Question

A rectangular garden, with length four times its width, is to be expanded so that both sides are increased by 6 yards. Let x represent the original width of the garden. Write an expression that models the area of the expanded garden.

Answer 1

Area of a rectangle

Initial explanation

We have that the area of a rectangle is given by:

area = side 1 * side 2

Then, for the original area

side 1 = length

side 2 = width

Then for the original area:

area = length * width

And for the new area:

area = new length * new width

Writing an expression for the new measures

We know that

x = width.

We know that the original length is 4 times the original width.

It is to say that the original length is 4 times x:

length = 4x

For the new measures

Since both sides are 6 yards bigger, then the new width is the original plus 6:

new width = width + 6 = x + 6

and the new length is the original length plus 6:

new length = length + 6 = 4x + 6

New area

Then, the new area:

area = new length * new width

Is given by:

area = (x + 6) (4x + 6)

= 4x² + 30x + 36

Answer: the area of the expanded garden is given by the expression

(x + 6) (4x + 6) = 4x² + 30x + 36