# A rectangular garden has a perimeter of 48 cm and an area of 140 sq. cm. What is the length of this garden?

Jul 27, 2016

Length of garden is $14$

#### Explanation:

Let the length be $L$ cm. and as area is $140$ cm., it being a product of length and width, width should be $\frac{140}{L}$.

Hence, perimeter is $2 \times \left(L + \frac{140}{L}\right)$, but as perimeter is $48$, we have

$2 \left(L + \frac{140}{L}\right) = 48$ or $L + \frac{140}{L} = \frac{48}{2} = 24$

Hence multiplying each term by $L$, we get

${L}^{2} + 140 = 24 L$ or ${L}^{2} - 24 L + 140 = 0$ or

${L}^{2} - 14 L - 10 L + 140 = 0$ or

$L \left(L - 14\right) - 10 \left(L - 14\right) = 0$ or

$\left(L - 14\right) \left(L - 10\right) = 0$

i.e. $L = 14$ or $10$.

Hence, dimensions of garden are $14$ and $10$ and length is more than width, it is $14$

Jul 27, 2016

The garden has sides of 14cm and 10cm. Length is 14cm.

#### Explanation:

We know that it is a rectangle, so each pair of opposite sides are the same length. We denote one set of sides length $x$ and the other set length $y$.

Therefore, the perimeter is given by $2 x + 2 y$.

$\therefore 2 x + 2 y = 48 c m$

The area of a rectangle is given by the product of it's length and breadth, ie

$A = x y = 140 c {m}^{2}$

$\implies x = \frac{140}{y}$

$2 \left(\frac{140}{y}\right) + 2 y = 48$

$\frac{280}{y} + 2 y = 48$

$140 + {y}^{2} = 24 y$

${y}^{2} - 24 y + 140 = 0$

$y = \frac{24 \pm \sqrt{{24}^{2} - 4 \left(1\right) \left(140\right)}}{2} = \frac{24 \pm \sqrt{16}}{2} = 10 \mathmr{and} 14$
$y = 10 \implies x = 14$
$y = 14 \implies x = 10$