Circular Arc Problems, 2

Category: Plane Geometry, Physics

"Published in Vacaville, California, USA"

How many revolutions will a car wheel of diameter 28 in. make over a period of half an hour if the car is traveling at 60 mi/hr?

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

This is a good application of circular arc problems in getting the number of revolutions of a car wheel especially when you're driving a car. In this problem, the distance traveled is not given but the speed or velocity and time are given. If you know the speed and time,then you can calculate the distance traveled as follows

where S is the distance traveled, V is the speed or velocity, and t is the travel time. Substitute the values of V and t, we have

Since the radius of a car wheel is expressed in inches, then we have to convert the distance traveled by car in inches as follows

Finally, we can get the number of revolutions of a car wheel as follows

where S is the total distance traveled or total length of a circular arc, R is the radius of a circle, and θ is the total angle of a circular arc in radians. Radians is a unit less value of an angle. 

Substitute the values of S and R in order to solve for the value of total angle, we have

Therefore, the number of revolutions of a car wheel is

Circular Arc Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

Los Angeles and New York are 2450 mi apart. Find the angle that the arc between these two cities subtends at the center of the earth. (The radius of the earth is 3960 mi.)

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

This is a good application of circular arc problems in getting the distance of two places. If you know the position of a city or a place like latitude and longitude, then you can calculate the central angle of two cities or places using the principles in solving spherical triangles. After the calculation of central angle, the distance of two places can be calculated. Since the distance of two cities is given in the problem, then we can proceed in calculating the central angle of two cities as follows

where S is the length of circular arc, R is the radius of a circle, and θ is the central angle of circular arc in radians. Radians is a unit less value of an angle.

Substitute the values of S and R in order to solve for the value of central angle as follows

You can also express the value of central angle in degrees as follows

or

 

Circular Sector Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

A sector in a circle of radius 25 ft has an area of 125 ft². Find the central angle of the sector.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The area of a circular sector is given by the formula

where A is the area of a circular sector, R is the radius of a circular sector, and θ is the central angle of circular sector in radians. Radians is a unit less value of an angle.

Substitute the values of A and R in order to solve for the value of central angle, as follows

You can also express the value of central angle in degrees, as follows

or