Question

Dimension of a Garden A homeowner wishes to fence in three adjoining garden plots, one for each of her children, as shown in the figure. If each plot is to be $80{\text{ft}}^2$in area and she has $88\text{ft}$of fencing material at hand, what dimensions should each plot have?

Short answer

The dimension of each plot is$12\times 6.66$ or $10\times 8$.

Step-by-step solution

Step 1. Write the given information.

The area of the plot is $80{\text{ft}}^2$.

The material to fence the plot is $88\text{ft}$.

Step 2. Form the system of equations.

Let the width of the plot be w and length be l. So, the area of the plot is,

$\begin{array}{c}wl=80\\ l=\frac{80}{w}\mathrm{...}\left(1\right)\end{array}$

The garden is divided into three adjoining plots, so the width of the plot need fence the plot is,

$\begin{array}{c}4w+6l=88\\ 4w+6\left(\frac{80}{w}\right)=88\\ \frac{4w^2+480}{w}=88\\ 4w^2+480=88w\end{array}$

Step 3. Solve for w.

Factorize the above quadratic equation and solve for w.

$\begin{array}{c}4w^2-88w+480=0\\ w^2-22w+120=0\\ w^2-12w-10w-120=0\\ \left(w-12\right)\left(w-10\right)=0\end{array}$

Apply zero product property.

$\begin{array}{c}w-12=0\\ w=12\end{array}$

And,

$\begin{array}{c}w-10=0\\ w=10\end{array}$

Substitute$12$ for w in equation (1).

$\begin{array}{c}l=\frac{80}{12}\\ =6.66\text{cm}\end{array}$

Substitute$10$ for w in equation (1).

$\begin{array}{c}l=\frac{80}{10}\\ =8\text{cm}\end{array}$

Thus, the dimension of each plot is$12\times 6.66$ or $10\times 8$.