Question

1. find the radius of the circle in the figure to the right.

the radius of the circle is
(type an integer or a simplified fraction.)

1. find the area of a sector of a circle having radius r and central angle θ.

r = 11.3 cm, θ = 81°
the area is approximately cm²
(do not round until the final answer. then round to the nearest tenth as needed.)

1. use the formula ω = \\( \frac { θ } { t } \\) to find the value of the missing variable.

θ = \\( \frac { 3 π } { 4 } \\) radians, t = 10 sec
ω = radians per sec
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using π as needed.)

1. the pulley shown has a radius of 13.58 cm. suppose it takes 12 sec for 63 cm of belt to go around the pulley.

(a) find the linear speed of the belt in centimeters per second.
(b) find the angular speed of the pulley in radians per second.
(a) the linear speed of the roller is cm per second.
(type an integer or decimal rounded to the nearest tenth as needed.)
(b) the angular speed of the roller is radian(s) per second.
(type an integer or decimal rounded to the nearest hundredth as needed. use the answer from part (a) to find this answer.)

Explanation:

Step1: Recall the arc - length formula

The arc - length formula is \(s = r\theta\), where \(s\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.
From the figure, \(s = 5\pi\) and \(\theta=\frac{3\pi}{4}\)

Step2: Solve for \(r\)

Substitute the values of \(s\) and \(\theta\) into the formula \(s = r\theta\). We get \(5\pi=r\times\frac{3\pi}{4}\).
To solve for \(r\), we can use the property of equality. Multiply both sides of the equation by \(\frac{4}{3\pi}\):

$$r=\frac{5\pi\times4}{3\pi}$$

Cancel out the \(\pi\) terms: \(r = \frac{20}{3}\)

Answer:

\(\frac{20}{3}\)