Question

a circle is inscribed in a square. find the circles area as a function of the perimeter of the square.
a ( f(x)=\frac{pi x^{2}}{64})
b ( f(x)=\frac{pi x^{2}}{16})
c ( f(x)=\frac{pi x^{2}}{4})
d ( f(x)=pi x^{2})

Explanation:

Step1: Let the perimeter of the square be $x$.

If the side - length of the square is $s$, then the perimeter of the square $x = 4s$, so $s=\frac{x}{4}$.

Step2: Find the radius of the inscribed circle.

Since a circle is inscribed in the square, the diameter of the circle is equal to the side - length of the square. Let the radius of the circle be $r$. Then $2r = s$, and substituting $s=\frac{x}{4}$, we get $r=\frac{s}{2}=\frac{x}{8}$.

Step3: Calculate the area of the circle.

The area formula of a circle is $A=\pi r^{2}$. Substitute $r = \frac{x}{8}$ into the formula: $A=\pi(\frac{x}{8})^{2}=\frac{\pi x^{2}}{64}$.

Answer:

A. $f(x)=\frac{\pi x^{2}}{64}$