Question

A small garden measures 9 ft by 13 ft. A uniform border around the garden increases the total area to 192 ft square. What is the width of the border ​

Answer 1

The width of border is 1.5 feet

Solution:

Given that small garden measures 9 ft by 13 ft

A uniform border around the garden increases the total area to 192 ft square

To find: width of border

Let the width of border be "x"

Then the new measures of garden are (9 + 2x) feet and (13 + 2x) feet

The total area of garden = 192 square feet

$(9 + 2x)(13 + 2x) =192$

$117 + 18x + 26x + 4x^2 = 192\\\\4x^2 + 44x - 75 = 0$

Let us solve the above equation by quadratic formula

$\text {For a quadratic equation } a x^{2}+b x+c=0, \text { where } a \neq 0\\\\x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$

Using the above formula,

For $4x^2 + 44x - 75 = 0$ , we have a = 4 ; b = 44 ; c = -75

Substituting the values of a = 4 ; b = 44 ; c = -75 in above quadratic formula we get,

$\begin{aligned}&x=\frac{-44 \pm \sqrt{44^{2}-4(4)(-75)}}{2 \times 4}\\\\&x=\frac{-44 \pm \sqrt{1936-1200}}{8}\\\\&x=\frac{-44 \pm \sqrt{3136}}{8}\\\\&x=\frac{-44 \pm 56}{8}\end{aligned}$

$\begin{aligned}&x=\frac{-44+56}{8} \text { or } x=\frac{-44-56}{8}\\\\&x=\frac{12}{8} \text { or } x=\frac{-100}{8}\\\\&x=1.5 \text { or } x=-12.5\end{aligned}$

Since "x" cannot be negative,

we get x = 1.5

Thus width of border is 1.5 feet