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

Answer:

$PQ = x + 13$

The dimension of the garden is 52.5ft by 43.5ft

Step-by-step 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

$\frac{8x}{8} = \frac{348}{8}$

$x = \frac{348}{8}$

$x =43.5$

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