Chapter 1.7, Problem 119E

Fencing a Garden A determined gardener has 120 ft of deer-resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area that is enclosed to be at least 800 ft². What range of values is possible for the length of her garden?

To determine

The range of values possible for the length of garden for fencing.

Answer to Problem 119E

The range of length of the garden for fencing is from $\underset{\_}{\text{20 ft to 40 ft}}$.

Explanation of Solution

The length of deer resistance fence is $120\text{ ft}$ and the area of the rectangular vegetable garden is atleast $800{\text{ft}}^2$.

Recall the fact that the value of perimeter of the garden will be equal to the length of the fence. That is, $P=120\text{ ft, }A=800{\text{ ft}}^2$.

Since the area of the rectangular vegetable garden is atleast $800{\text{ft}}^2$, $A\ge 800{\text{ ft}}^2$.

Note that, the perimeter of the rectangular vegetable garden is $P=2\left(l+b\right)$ and the Area of the rectangular vegetable garden is $A=\left(l\times b\right)$ where l is the length and b breadth of the garden.

Substitute $P=120\text{ ft}$ in $P=2\left(l+b\right)$:

$\begin{array}{l}2\left(l+b\right)=120\\ \left(l+b\right)=\frac{120}{2}\\ l+b=60\\ b=60-l\end{array}$

Substitute $A\ge 800{\text{ ft}}^2$ in $A=\left(l\times b\right)$:

$l\times b\ge 800$

Substitute $b=60-l$ in $l\times b\ge 800$ as follows:

$\begin{array}{l}l\times \left(60-l\right)\ge 800\\ 60l-l^2\ge 800\\ l^2-60l\le -800\left[\text{Multiplying by }-1\right]\\ l^2-60l+{\left(-\frac{60}{2}\right)}^2\le -800+{\left(-\frac{60}{2}\right)}^2\\ l^2-60l+900\le -800+900\end{array}$

Simplify further as follows:

$\begin{array}{l}{l}^2-60l+900\le -800+900\\ {\left(l-30\right)}^2\le 100\\ \sqrt{{\left(l-30\right)}^2}\le \sqrt{100}\\ \left|l-30\right|\le 10\\ -10\le l-30\le 10\end{array}$

Solve further as follows:

$\begin{array}{l}-10+30\le l\le 10+30\\ 20\le l\le 40\end{array}$

Thus, the range of length of the garden for fencing is from $\underset{\_}{\text{20 ft to 40 ft}}$.