A gardener has 80 m of fencing wire. He can use this wire to enclose either a circular lawn or a square lawn. If he uses the entire wire in each case, what is the approximate ratio of the area of the circular lawn to the area of the square lawn?

Correct Option: 1.27 : 1

To find the ratio of the area of the circular lawn to the area of the square lawn, let us analyze both cases step-by-step using the given perimeter (total length of fencing wire, $L=80\text{m}$).

Step 1: Calculate the area of the circular lawn
The total length of the wire forms the circumference of the circle.
Circumference $C=2\pi r=80$

Solving for the radius $r$:

$r=\frac{80}{2\pi }=\frac{40}{\pi }$

Now, compute the area of the circular lawn ($A_{\text{circle}}$):

$A_{\text{circle}}=\pi r^2=\pi {\left(\frac{40}{\pi }\right)}^2=\frac{1600}{\pi }$

Step 2: Calculate the area of the square lawn
The total length of the wire forms the perimeter of the square.
Perimeter $P=4s=80$

Solving for the side length $s$:

$s=\frac{80}{4}=20\text{m}$

Now, compute the area of the square lawn ($A_{\text{square}}$):

$A_{\text{square}}=s^2={20}^2=400$

Step 3: Find the ratio of the areas
Ratio = $\frac{A_{\text{circle}}}{A_{\text{square}}}=\frac{1600/\pi }{400}=\frac{4}{\pi }$

Using the approximate value of $\pi \approx 3.14159$:

$\text{Ratio}=\frac{4}{3.14159}\approx 1.273$

Thus, the approximate ratio of the area of the circular lawn to the area of the square lawn is 1.27 : 1.