Question

name date block im2 cl m2t3 individual test (a) show all work on all problems for full credit. be sure to write the formula used and show any substitutions and work for credit. write your final answer in the blanks provided. dont forget proper units when available. 1a) calculate the length of \\( \widehat { a b } \\) in terms of simplest fraction \\( \pi \\) form. \\( \widehat { a b } = \\) inch 1b) calculate the area of the sector. leave in simplest fraction \\( \pi \\) form. \\( a = \\)

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(l\) of a circle is \(l=\frac{\theta}{360}\times2\pi r\), where \(\theta\) is the central angle and \(r\) is the radius of the circle. Given \(r = 10\) inches and \(\theta=60^{\circ}\).
Substitute \(\theta = 60\) and \(r = 10\) into the formula: \(l=\frac{60}{360}\times2\pi\times10\).

Step2: Simplify the arc - length formula

Simplify \(\frac{60}{360}=\frac{1}{6}\). Then \(l=\frac{1}{6}\times20\pi=\frac{10\pi}{3}\) inches.

Step3: Recall the sector - area formula

The formula for the area \(A\) of a sector of a circle is \(A=\frac{\theta}{360}\times\pi r^{2}\), where \(\theta\) is the central angle and \(r\) is the radius of the circle.
Substitute \(\theta = 60\) and \(r = 10\) into the formula: \(A=\frac{60}{360}\times\pi\times10^{2}\).

Step4: Simplify the sector - area formula

Simplify \(\frac{60}{360}=\frac{1}{6}\). Then \(A=\frac{1}{6}\times100\pi=\frac{50\pi}{3}\) square inches.

Answer:

1a) \(\frac{10\pi}{3}\) inches
1b) \(\frac{50\pi}{3}\) square inches