Question

a. find the length of a 15° arc of a circle whose radius is 4 cm.
b. explain why the arc length in part (a) is longer or shorter in centimeters if the radius had been 7 cm.
c. find the arc length when the radius is increased to 7 cm.
a. the length of the 15° arc is \\( \frac{\pi}{3} \\) cm
(simplify your answer. type an exact answer, using \\( \pi \\) as needed. use integers or fractions for any numbers in the expression )
b. explain why the arc length in part (a) is longer or shorter in centimeters if the radius had been 7 cm.
the arc has to be because 7 cm is cm, and the radius is in the of the computation

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) of a circle is \(s = r\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians. First, convert the angle from degrees to radians. We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees). For \(\theta = 15^{\circ}\), \(\theta=\frac{15\pi}{180}=\frac{\pi}{12}\) radians.

Step2: Calculate the arc - length for \(r = 4\) cm

Using the formula \(s=r\theta\), with \(r = 4\) cm and \(\theta=\frac{\pi}{12}\) radians, we have \(s = 4\times\frac{\pi}{12}=\frac{\pi}{3}\) cm.

Step3: Analyze the effect of radius change on arc - length (part b)

The arc - length formula \(s=r\theta\) shows a direct - proportional relationship between \(s\) and \(r\) (since \(\theta=\frac{\pi}{12}\) is constant). When \(r\) increases from \(4\) cm to \(7\) cm (\(7>4\)), and \(\theta=\frac{\pi}{12}\) (constant), since \(s\) and \(r\) are directly proportional (\(s = r\theta\) with \(\theta\) constant), the arc has to be longer because \(7\) cm is greater than \(4\) cm, and the radius is in the numerator of the arc - length formula \(s=r\theta\).

Step4: Calculate the arc - length for \(r = 7\) cm

Using the formula \(s=r\theta\) with \(r = 7\) cm and \(\theta=\frac{\pi}{12}\) radians. Then \(s=7\times\frac{\pi}{12}=\frac{7\pi}{12}\) cm.

Answer:

a. \(\frac{\pi}{3}\) cm
b. The arc has to be longer because \(7\) cm is greater than \(4\) cm, and the radius is in the numerator of the computation.
c. \(\frac{7\pi}{12}\) cm