Respuesta :
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
[tex](9 + 2x)(13 + 2x) =192[/tex]
[tex]117 + 18x + 26x + 4x^2 = 192\\\\4x^2 + 44x - 75 = 0[/tex]
Let us solve the above equation by quadratic formula
[tex]\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}[/tex]
Using the above formula,
For [tex]4x^2 + 44x - 75 = 0[/tex] , we have a = 4 ; b = 44 ; c = -75
Substituting the values of a = 4 ; b = 44 ; c = -75 in above quadratic formula we get,
[tex]\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}[/tex]
[tex]\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}[/tex]
Since "x" cannot be negative,
we get x = 1.5
Thus width of border is 1.5 feet
