Question

A gardener wants to create a rectangular vegetable garden in a backyard. He wants it to have a total area of 112 square feet, and it should be 14 feet longer than it is wide. What dimensions should he use for the vegetable garden? Round to the nearest hundredth of a foot.

Answer 1

The rectangular garden is going to have an area of 112 square feet.

However, it should be 14 feet longer than it is wide. This means, if it is w feet wide, then it should be (w + 14) feet long. Hence we now have;

`\begin{gathered} \text{Area}=l* w \\ 112=(w+14)* w \\ 112=w^2+14w \\ \text{Move all terms to one side of the equation and we now have;} \\ w^2+14w-112=0 \\ We\text{ can use the quadratic equation formula to solve, as follows;} \\ x=\frac{-b(+)/(-)\sqrt[]{b^2-4ac}}{2a} \\ a=1,b=14,c=-112 \\ w=\frac{-14(+)/(-)\sqrt[]{14^2-4(1)(-112)}}{2(1)} \\ w=\frac{-14(+)/(-)\sqrt[]{196+448}}{2} \\ w=\frac{-14(+)/(-)\sqrt[]{644}}{2} \\ w=(-14(+)/(-)25.3772)/(2) \\ w=(-14+25.3772)/(2)\text{ OR} \\ w=(-14-25.3772)/(2) \\ w=(11.3772)/(2)\text{ OR} \\ w=(-39.3772)/(2) \\ w=5.6886,OR,w=-19.6886 \\ \text{Knowing that the dimension can only be a positive value, we will take} \\ w=5.69\text{ (rounded to the nearest hundredth)} \\ \text{If the length is 14 feet longer than the width,} \\ \text{Then the length would be 5.69+14} \\ \text{Length would be 19.69 feet} \end{gathered}`

Therefore, the length is 19.69 feet, and the width is 5.69 feet