Question

The lengths of a rectangular garden, the inner rectangle, is 9ft more than its width. For the garden. Let x = width, x + 9 = length. It is surrounded by a brick walkway 4ft wide. Suppose the total area of the walkway is 400ft^2.

a. Write a polynomial to represent the length of PQ

b. What are the dimensions of the garden?

Answer 1

`PQ = x + 13`

The dimension of the garden is 52.5ft by 43.5ft

Explanation:

Given

The shape above

Required

Determine PQ

Determine dimension of the garden

Calculating Length PQ

Represent the length of the inner rectangle as L

Represent the width of the inner rectangle as W

`W = x`

`L = x + 9`

The distance between the inner rectangle and the outer triangle is 4ft

This implies that;

`PQ = L + 4`

Substitute
`x + 9` for
`L`

`PQ = x + 9 + 4`

`PQ = x + 13`

Also;

`QR = W + 4`

Substitute
`x` for
`W`

`QR = x + 4`

Calculating The Dimension of The Garden

First, we need to determine the Area of the inner rectangle

`Area_1 = L * W`

Recall that
`W = x` and
`L = x + 9`

So;

`Area_1 = x * (x + 9)`

For the bigger rectangle

`Area_2 = PQ * QR`

Recall that
`PQ = x + 13` and
`QR = x + 4`

So;

`Area_2 = (x + 13)(x + 4)`

Given that the Area of the walkway is
`400ft^2`

This implies that

`Area_2 = Area_1 + 400`

Substitute
`Area_1 = x * (x + 9)` and
`Area_2 = (x + 13)(x + 4)`

`(x + 13)(x + 4) = x * (x + 9) + 400`

Open All Brackets

`x^2 + 13x + 4x + 52 = x^2 + 9x + 400`

`x^2 + 17x + 52 = x^2 + 9x + 400`

Collect Like Terms

`x^2 - x^2 + 17x - 9x = 400- 52`

`8x = 348`

Divide both sides by 8

`(8x)/(8) = (348)/(8)`

`x = (348)/(8)`

`x =43.5`

Since the dimensions of the garden is
`W = x` and
`L = x + 9`

Substitute 43.5 for x in both cases

`L = 43.5 + 9 = 52.5`

`W = 43.5`

Hence, the dimension of the garden is 52.5ft by 43.5ft