Question

in the chart below, the a (area) of a circle is given. find the values of the r (radius), d (diameter), and c (circumference) for this particular circle. (simplify your answers. type exact answers, using π as needed.) a = \frac{144pi}{yd^{2}}

Explanation:

Step1: Recall the area formula for a circle

The area formula of a circle is $A = \pi r^{2}$. Given $A=\frac{\pi}{144}\text{ yd}^2$. So, $\pi r^{2}=\frac{\pi}{144}$.

Step2: Solve for the radius $r$

Divide both sides of the equation $\pi r^{2}=\frac{\pi}{144}$ by $\pi$ to get $r^{2}=\frac{1}{144}$. Then take the square - root of both sides. Since $r>0$ (radius is a non - negative quantity), $r = \frac{1}{12}\text{ yd}$.

Step3: Calculate the diameter $d$

The diameter $d$ of a circle is related to the radius by $d = 2r$. Substitute $r=\frac{1}{12}\text{ yd}$ into the formula, so $d=2\times\frac{1}{12}=\frac{1}{6}\text{ yd}$.

Step4: Calculate the circumference $C$

The circumference formula of a circle is $C = 2\pi r$ or $C=\pi d$. Using $r=\frac{1}{12}\text{ yd}$, we have $C = 2\pi\times\frac{1}{12}=\frac{\pi}{6}\text{ yd}$.

Answer:

$r=\frac{1}{12}\text{ yd}$, $d = \frac{1}{6}\text{ yd}$, $C=\frac{\pi}{6}\text{ yd}$