Question

1. the circles have a radius of 1, 2, 3, and 4 units, respectively. the diameter divides each circle into two equal sectors. find the arc length of one half of each of these circles. then fill in the table with the indicated values.
2. why does the arc length increase as the radius increases, but the ratio of arc length to the radius remains the same?

Explanation:

Step1: Recall the formula for arc length

The formula for arc length \(L = r\theta\). For a semic - circle, the central angle \(\theta=\pi\) radians.

Step2: Calculate arc length for \(r = 1\)

When \(r = 1\), using \(L=r\theta\) with \(\theta=\pi\), we have \(L = 1\times\pi=\pi\). The ratio \(\frac{L}{r}=\frac{\pi}{1}=\pi\).

Step3: Calculate arc length for \(r = 2\)

When \(r = 2\), using \(L=r\theta\) with \(\theta=\pi\), we have \(L = 2\times\pi = 2\pi\). The ratio \(\frac{L}{r}=\frac{2\pi}{2}=\pi\).

Step4: Calculate arc length for \(r = 3\)

When \(r = 3\), using \(L=r\theta\) with \(\theta=\pi\), we have \(L=3\times\pi = 3\pi\). The ratio \(\frac{L}{r}=\frac{3\pi}{3}=\pi\).

Step5: Calculate arc length for \(r = 4\)

When \(r = 4\), using \(L=r\theta\) with \(\theta=\pi\), we have \(L = 4\times\pi=4\pi\). The ratio \(\frac{L}{r}=\frac{4\pi}{4}=\pi\).

Answer:

Radius	Arc Length	\(\frac{\text{Arc Length}}{\text{Radius}}\)	Radian Measure of Central Angle
\(2\)	\(2\pi\)	\(\pi\)	\(\pi\)
\(3\)	\(3\pi\)	\(\pi\)	\(\pi\)
\(4\)	\(4\pi\)	\(\pi\)	\(\pi\)

For question 3:
The arc length formula is \(L = r\theta\). When \(\theta\) (the radian measure of the central angle, which is \(\pi\) for a semi - circle) is constant, \(L\) is a linear function of \(r\) (\(L=\theta\times r\), and since \(\theta=\pi\) (constant for a semi - circle), as \(r\) increases, \(L\) increases. The ratio \(\frac{L}{r}=\theta\), and since \(\theta=\pi\) (constant for a semi - circle), the ratio \(\frac{\text{Arc Length}}{\text{Radius}}\) remains the same.