Conestoga wagons in the 1800s had wheels that were smaller in the front than in the back. The front wheels had . The top of the front wheel measured from the ground. The rear wheels had . The top of the rear wheel measured inches from the ground.

(For these problems, disregard the hub at the center of the wheel. Assume the spokes meet in the center at a point. Leave in your answers.)

### 1.

Find the area and the circumference of each wheel.

#### a.

Front-wheel area:

#### b.

Front-wheel circumference:

Rear-wheel area:

#### d.

Rear-wheel circumference:

### 2.

Determine the central angle between the spokes on each wheel.

Front wheel:

Rear wheel:

### 3.

Find the distance on the circumference between two consecutive spokes for each wheel.

Front wheel:

Rear wheel:

### 4.

The wagons could cover a distance of per day. How many more times would the front wheel turn than the back wheel on an average day?

### 5.

A wheel rotates per minute. Write an equation for how many degrees it rotates in .

## Set

A right triangle has been drawn from a point on a circle centered at The angle of rotation has been drawn between the initial ray and the terminal ray.

• Write the value of as defined in today’s task. (You will need to find the radius.)

• Use right angle trigonometry to solve for angle inside the triangle.

• Use the measure of to find (the angle of rotation).

### 12.

In each graph, the angle of rotation is indicated by an arc and . Describe the angles of rotation that make the -values of the points positive and the angles of rotation that make the -values negative.

### 13.

What do you notice about the -values and the value of cosine in the graphs?

### 14.

In the graph, the radius of the circle is . The intersections of the circle and the axes are labeled. Based on your observation in #11, what do you think the value of cosine might be for the following values of ?

## Go

Perform the indicated operation and divide out any common factors in your answers.