Question

consider circle o, in which arc xy measures $16\pi$ cm. the length of a radius of the circle is 32 cm. what is the circumference of the circle? $\pi$ units is the ratio of the arc length to the circumference? 32 50 is the measure of central angle xoy? 64

Explanation:

First Sub - Question: Circumference of the circle

Step1: Recall the formula for the circumference of a circle

The formula for the circumference \(C\) of a circle with radius \(r\) is \(C = 2\pi r\).

Step2: Substitute the given radius into the formula

We are given that the radius \(r=32\) cm. Substituting \(r = 32\) into the formula \(C=2\pi r\), we get \(C = 2\pi\times32=64\pi\) cm.

Step1: Identify the arc length and circumference

The arc length \(l = 16\pi\) cm and the circumference \(C=64\pi\) cm (from the first sub - question).

Step2: Calculate the ratio

The ratio of the arc length to the circumference is \(\frac{l}{C}=\frac{16\pi}{64\pi}\). The \(\pi\) terms cancel out, and \(\frac{16}{64}=\frac{1}{4}\).

Step1: Recall the relationship between arc length, circumference, and central angle

The ratio of the arc length to the circumference is equal to the ratio of the central angle (in degrees) to \(360^{\circ}\). Let the central angle be \(\theta\). So, \(\frac{l}{C}=\frac{\theta}{360^{\circ}}\).

Step2: Substitute the known ratio and solve for \(\theta\)

We know that \(\frac{l}{C}=\frac{1}{4}\) (from the second sub - question). So, \(\frac{\theta}{360^{\circ}}=\frac{1}{4}\). Cross - multiplying gives \(\theta=\frac{360^{\circ}}{4} = 90^{\circ}\).

Answer:

The circumference of the circle is \(64\pi\) cm.

Second Sub - Question: Ratio of arc length to circumference