Question

solving equations for circles
homework
directions: show all work, starting with the formula.

1. **a circle has a circumference of 37.68 cm. find its diameter.

area = \\(\pi r^2\\)
circumference = \\(\pi d\\)

1. **a circle has an area of 100\\(\pi\\) square centimeters. find its radius.
2. **a circle has a radius of 12.56 miles. find its area.

Explanation:

Problem 1:

Step1: Recall the formula for circumference

The formula for the circumference \( C \) of a circle is \( C=\pi d \), where \( d \) is the diameter. We know \( C = 37.68\) cm and we take \( \pi\approx3.14 \).

Step2: Solve for diameter \( d \)

From \( C=\pi d \), we can re - arrange the formula to \( d=\frac{C}{\pi} \). Substitute \( C = 37.68\) cm and \( \pi = 3.14\) into the formula: \( d=\frac{37.68}{3.14}=12 \) cm.

Step1: Recall the formula for the area of a circle

The formula for the area \( A \) of a circle is \( A=\pi r^{2} \), where \( r \) is the radius. We know that \( A = 100\pi\) square centimeters.

Step2: Solve for the radius \( r \)

Substitute \( A = 100\pi\) into the formula \( A=\pi r^{2} \). We get \( 100\pi=\pi r^{2} \). Divide both sides of the equation by \( \pi \), we have \( 100=r^{2} \). Taking the square root of both sides, \( r = 10\) (we consider the positive square root since radius is a non - negative quantity) centimeters.

Step1: Recall the formula for the area of a circle

The formula for the area \( A \) of a circle is \( A=\pi r^{2} \), where \( r \) is the radius. We know that \( r = 12.56\) miles and we take \( \pi\approx3.14 \).

Step2: Calculate the area

Substitute \( r = 12.56\) miles and \( \pi = 3.14\) into the formula \( A=\pi r^{2} \). So \( A=3.14\times(12.56)^{2}=3.14\times157.7536 = 495.346304\) square miles.

Answer:

The diameter of the circle is 12 cm.

Problem 2: