Problem Solving Involving Angles

By Georgette April D. Policarpio

Definition of Terms

Intercepted Arc - arc length formed by the rays

of the central angle and the section of the circle.

- It is the rate of movement of a particle moving in constant speed along a circular arc of radius r.

- It is given by ratio of the arc length over time.

Linear Speed

– the rate of movement in which the radius r forms an angle to correspond with the arc length s.

Angular Speed

The Length of a Circular Arc

• The lengths of the arc intercepted by the central angle is given by:

Where: r is the radius of the circleθ is the non negative radian measure

of a central angle of the circle.

rs

Examples

• A circle has a radius of 20 centimeters. Find the length of the arc intercepted by a central angle of 120°.

Answer: 41.89 cm.

• The intercepted arc of an angle that measures 120° is equal to 42 centimeters. Find the measure of the radius of the circle.

Examples

• The length of the intercepted arc of a circle with a radius of 20 centimeters is 41.89 centimeters. Find the measure of the intercepted angle.

Examples

Linear Speed

the linear speed of the particle is given by:

Where:s is the length of the arc traveledt is the length of time

ts

v

Example

• The second hand of a clock is 10.5 centimeters long. Find the linear speed of the tip of this second hand as it passes around the clock face.

Angular Speed

The angular speed of the particle is given by:

Where:θ is the angle in radians

corresponding to an arc length s, t is the length of time

t

• Consider the motion of a point P along a circle of radius 5 cm. Starting at A(5,0).

P

0 A(5,0)

1. Suppose that OP after 10 seconds, the

angle covered by OP is .radians4

5

Solve for the angular speed in radians per second.

Answer: second per radians 8

Linear Speed In Terms Of Angular Speed

Formula:

Where:r is the radius w is the angular speed

rv

Examples

1. A wheel with a diameter 12 inches is rotating at 340 revolutions per minute. Find the angular speed in radians per second and the linear speed of a point in the wheel in terms of meter per second.

• For angular speed:

• For linear speed:

334

secs. 60min. 1

rev 1rad 2

min.rev. 340

smm 4618515340

334

..

2. A car with a tire with a radius of 16 inches is rotating at 450 revolutions per minute.

a. the angular speed in radians per second.

b. Linear speed of a point on the rim in meters per second.

c. The measure α of the angle generated by a spoke of the wheel in a 30 seconds.

Answer key for the examples:a. ω = 3π/2b. v = 1.9151 m/s

Note: get the linear speed with the use of the angular speed and and the radius.

c. α = 45π radNote: Multiply the angular speed by the length of time t to get the measure of the angle formed.

Exercises

Give the radian measure of the central angle of a circle that intercepts an arc of length s.• r = 16 inches; s = 4 ft.• r = 8 yrds; s = 18 yrds.• r = 10 inches; s = 40 inches

Exercises

Find the length of the arc on a circle of radius r intercepted by the central angle θ.• r = 12 inches; θ = 45°• r = 8 feet; θ = 225°• r = 9 yards; θ = 315°

Exercises

Solve the following Word Problems.1. A wheel is rotating at 250 revolutions per

minute. Find the angular speed in radians per second.

2. A car with a tire diameter of 32 inches is rotating at 450 revolutions per minute. Find the speed of the car to the nearest kilometer per hour.

4. The radius of a wheel is 80 cm. If the wheel rotates through an angle of 60°, how many centimeters does it move? Express your answer to two decimal places.

Exercises

5. Suppose the figure shows a highway sign that warns of a railway crossing. The lines that form the cross pass through the circles’ center and intersect at right angles.

Exercises

If the radius of the circle is 24 inches, Find the length of each of the four arcs formed by the cross. Express your final answer in the nearest hundredths.

Exercises

• Assuming Earth to be a sphere of radius 4000 miles, how many miles north of the equator is Miami, Florida, if it is 26 degrees north of the equator? Show your complete and organized computation.

Assignment