### All Precalculus Resources

## Example Questions

### Example Question #6 : Angular And Linear Velocity

A plane is traveling Northeast. If the eastern component of it's velocity is , how fast is the plane traveling?

**Possible Answers:**

None of the other answers

**Correct answer:**

Since the plane is traveling NE, we know that it is traveling at an angle of 45 degrees north of east. Therefore, we can represent this in a diagram:

We have all of the information we need to solve for the velocity of the plane. We have an angle and a side adjacent to that angle. Therefore, we can use the cosine function to solve for the velocity.

Rearranging for v, we get:

### Example Question #1 : Angular And Linear Velocity

A car is traveling NW at a speed of mph. What are the magnitudes of each of its North and West components?

**Possible Answers:**

North: mph

West: mph

North: mph

West: mph

North: mph

West: mph

North: mph

West: mph

North: mph

West: mph

**Correct answer:**

North: mph

West: mph

If the car is traveling NW, it means that the angle is degrees above the x-axis. Using this angle, and the fact that the magnitude of the velocity is mph, we can find the x and y components, or the west and north components using trigonometry.

### Example Question #8 : Angular And Linear Velocity

An object is moving in a circular path. How will its linear velocity change if the diameter of the circular path was decreased by one half?

**Possible Answers:**

**Correct answer:**

Write the formula for the linear velocity.

The diameter is twice the radius.

If the diameter was decreased by one half, the radius will also decrease by one half.

Therefore:

The linear velocity will decrease by one half.

### Example Question #9 : Angular And Linear Velocity

Find the linear velocity in meters per second of an object if it took hours to travel a distance of kilometers. Round to the nearest integer.

**Possible Answers:**

**Correct answer:**

Write the linear velocity formula.

Since the answer is required in meters per second, two conversions will need to be made.

Convert 30 kilometers to meters.

Convert 2 hours to seconds.

Apply the formula.

### Example Question #10 : Angular And Linear Velocity

The second hand of a clock is long. Find the linear speed of the end of this second hand.

**Possible Answers:**

None of the other answers.

**Correct answer:**

Linear velocity is defined as distance over a period of time. For instance if a person ran 1 mile or approximately 1600 meters in 7 minutes, the they would have covered about 230 meters per minute. Let's assume this person was running around a track. We could also measure their speed from a central angle and represent their speed as the amount of degrees (or radians) they ran around per unit time as well. This is considered angular speed. A perfect example of both are the hands on a clock. There is a relationship between arc length which we designate s, the radius r, and the angle (theta). The relationship is . So the length of the arc (s) is equal to the radius of the circle the arc is on and what section of the pie it covers ( this is akin to how many degrees our track runner ran "through" or around) . Thus, if we wanted the linear speed around a circle we could say or . Key measurements that you will need to know are how many degrees in a circle of which there are 360 or . Before you follow the step by step solution below, go back and see if you can use this new information to arrive at the correct answer.

The actual answer is .

To see why, note that the second hand spins around a total of 360 degrees or .

.

And how long does it take for the hand to go around? Linear speed of the clock second hand is

(rounded answer).

### Example Question #1 : Solve Linear Velocity Problems

A clock has a second hand measuring 12cm. What is the linear speed of the tip of the second hand?

**Possible Answers:**

None of these/

**Correct answer:**

Linear speed is equal to the arc length traversed divided by the time. We use s to denote arc length.

### Example Question #1 : Angles In The Coordinate Plane

Convert radians to degrees.

**Possible Answers:**

**Correct answer:**

Write the conversion factor between radians and degrees.

Cancel the radians unit by using dimensional analysis.

### Example Question #1 : Angles In The Coordinate Plane

Convert to degrees.

**Possible Answers:**

**Correct answer:**

Write the conversion factor of radians and degrees.

Substitute the degree measure into .

### Example Question #3 : Angles In The Coordinate Plane

Determine the angle in degres made in the plane by connecting a line segment from the origin to .

Assume

**Possible Answers:**

**Correct answer:**

Firstly, since the point is in the 3rd quadrant, it'll be between and . If we take to be the horizontal, we can form a triangle with base and leg of values and . Solving for the angle in the 3rd quadrant given by ,

Since this angle is made by assuming to be the horizontal, the total angle measure is going to be:

### Example Question #4 : Angles In The Coordinate Plane

Find the degree measure of radians. Round to the nearest integer.

**Possible Answers:**

**Correct answer:**

In order to solve for the degree measure from radians, replace the radians with 180 degrees.

The nearest degree is .