# The perimeter of a rectangular flower garden is 60m and its area is 225 m^2. How do you find the length of the garden?

Mar 7, 2018

Length of the garden = 15m

#### Explanation:

Let $l$ be the length (longer side) of the rectangular garden, and $w$ be the width(shorter side).

Perimeter = $2 \left(l + w\right) = 60 m$-----(1) given.
Area = $l \times w = 225 {m}^{2}$------(2) given

(1) $\implies 2 l + 2 w = 60$

$\implies 2 l = 60 - 2 w$

$\implies l = \frac{60 - 2 w}{2}$

Substitute $l$ in (2):

$\implies w \times \frac{60 - 2 w}{2} = 225$

$\implies 60 w - 2 {w}^{2} = 225 \times 2$

$\implies - 2 {w}^{2} + 60 w - 550 = 0$

$\implies 2 {w}^{2} - 60 w + 550 = 0$

$\implies {w}^{2} - 30 w + 225 = 0$

$\implies {w}^{2} - 15 w - 15 w + 225 = 0$

$\implies w \left(w - 15\right) - 15 \left(w - 15\right) = 0$

$\implies \left(w - 15\right) \left(w - 15\right) = 0$

$\implies w - 15 = 0$

$\implies \textcolor{red}{w = 15 m}$

So, the width is $w = 15 m$.

$\therefore$ (1) $\implies 2 \left(l + w\right) = 60$

$\implies 2 \left(l + 15\right) = 60$

$\implies 2 l + 30 = 60$

$\implies l = \frac{30}{2} = 15$

$\textcolor{red}{l = 15 m}$

That means the length of the rectangular garden is also $15 m$

This implies that the garden is $\textcolor{red}{\square}$ shaped with $l = 15 m$ and $w = 15 m$ i.e. each Side = $15 m$