Question

determine the area of a circular enclosure and a square enclosure made with the given amount of fence. compare the areas and comment. 72 meters of fence. the area of a square enclosure created with 72 meters of fence is about square meters. (do not round until the final answer. then round to the nearest whole number as needed.)

Explanation:

Step1: Find side - length of square

The perimeter of a square is $P = 4s$, where $P$ is the perimeter and $s$ is the side - length. Given $P=72$ meters, then $s=\frac{P}{4}=\frac{72}{4}=18$ meters.

Step2: Calculate area of square

The area of a square is $A_{square}=s^{2}$. Substituting $s = 18$ meters, we get $A_{square}=18^{2}=324$ square meters.

Step3: Find radius of circle

The circumference of a circle is $C = 2\pi r$, where $C$ is the circumference and $r$ is the radius. Given $C = 72$ meters, then $r=\frac{C}{2\pi}=\frac{72}{2\pi}=\frac{36}{\pi}$ meters.

Step4: Calculate area of circle

The area of a circle is $A_{circle}=\pi r^{2}$. Substituting $r=\frac{36}{\pi}$ meters, we have $A_{circle}=\pi\times(\frac{36}{\pi})^{2}=\pi\times\frac{1296}{\pi^{2}}=\frac{1296}{\pi}\approx\frac{1296}{3.14}\approx412.74$ square meters.

Step5: Compare areas

We see that $A_{circle}\approx413$ square meters and $A_{square}=324$ square meters. The area of the circular enclosure is larger than the area of the square enclosure.

Answer:

The area of the square enclosure is 324 square meters. The area of the circular enclosure is approximately 413 square meters. The circular enclosure has a larger area.