Question

A square lawn is bounded on three sides by a path of $4m$ wide. If the area of the path is $\frac{7}{8}$ that of the lawn, find the dimensions of the lawn.

Answer 1

Step 1. Finding the area of square lawn and path:

Let the length of the side of the square lawn be $x\mathrm{m}$.

The area of square lawn $=side\times side=x^2{\mathrm{m}}^2$
The area of the path $=(x+8)\times 4+x\times 4+x\times 4$

$=8x+4x+32$

$=12x+32m^2$
Step 2. Finding the dimensions of the lawn:

Now, the area of the path is $\frac{7}{8}$ that of the lawn.

$$\begin{gathered} 12x+32=\frac{7}{8}{x}^2 \\ 96x+256=7x^2 \\ 7x^2-96x-256=0 \\ 7x^2-112x+16x-256=0 \\ 7x(x-16)+16(x-16)=0 \\ (7x+16)(x-16)=0 \\ x=16m,-\frac{16}{7}m \end{gathered}$$

But the length cannot be negative.
Thus, the dimensions of the square lawn is $16m$.