# A rectangular garden 42 by 20 feet is surrounded by a walk way of uniform width. If the total area of the garden and walkway is 1248 square feet, what is the the width of the walkway?

Sep 10, 2015

I found $3 \text{ft}$

#### Explanation:

You can take the total area $1248 {\text{ft}}^{2}$ and subtract the area of the garden which is $42 \cdot 20 = 840 {\text{ft}}^{2}$ and get:
$1248 - 840 = 408 {\text{ft}}^{2}$
So:

Sep 10, 2015

The walkway is $3$ feet wide

#### Explanation:

Define
$\textcolor{w h i t e}{\text{XXX}} w$: width of walkway

Area of garden (including walkway)
$\textcolor{w h i t e}{\text{XXX}} \left(42 + 2 w\right) \times \left(20 + 2 w\right) = 1248$
With some basic algebra and arithmetic:
$\rightarrow \textcolor{w h i t e}{\text{XX}} 4 {w}^{2} + 124 w + 840 = 1248$
$\rightarrow \textcolor{w h i t e}{\text{XX}} 4 {w}^{2} + 124 w - 408 = 0$
$\rightarrow \textcolor{w h i t e}{\text{XX}} {w}^{2} + 31 w - 102 = 0$
$\rightarrow \textcolor{w h i t e}{\text{XX}} \left(w - 3\right) \left(w + 34\right) = 0$

and (since the width can't be a negative number)
$\rightarrow \textcolor{w h i t e}{\text{XX}} w = 3$