Question

the given diagram shows an angle measuring 0.89 radians with an initial ray pointing in the 3 - oclock position. there are three circles centered at the angles vertex with radii 4, 6, and 9 cm long creating three different terminal points.

a each terminal point has a horizontal distance to the right of the circles vertical diameters that measures 3 radii lengths. how can we compare the relative sizes of the blue arrows and the circles radius lengths?

b each terminal point has a horizontal distance to the right of the circles vertical diameters that is 88 percent of its corresponding circles radius. try again first, complete part (a). once you have the answer to part (a), how do you represent that comparison as a percent?

c if another circle was drawn with a radius 21 cm long, what would be the terminal points horizontal distance to the right of the vertical diameter in centimeters?

d the red arcs represent the arcs subtended by the angles rays for the three circles. each of these arcs is 0.89 times as long as the corresponding circles radius. try again start by drawing a diagram. you will need the answers to previous parts to help you.

correct! since the angle measures 0.89 radians, the angles rays subtend an arc that is 0.89 times as long as the radius for any circle centered at the angles vertex.

Explanation:

Step1: Recall the formula for arc length

The formula for arc length \(s = r\theta\), where \(r\) is the radius and \(\theta\) is the angle in radians. Given \(\theta=0.89\) radians.

Step2: Calculate the arc length for \(r = 21\) cm

Substitute \(r = 21\) cm and \(\theta = 0.89\) into the formula \(s=r\theta\). So \(s=21\times0.89\).

$$s = 21\times0.89=(20 + 1)\times0.89=20\times0.89+1\times0.89=17.8+0.89 = 18.69$$

Answer:

\(18.69\)