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Lesson 5 Moving Shadows Practice Understanding

Ready

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.

an image of a real wagon with 2 different sized wheels

(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:

c.

Rear-wheel area:

d.

Rear-wheel circumference:

2.

Determine the central angle between the spokes on each wheel.

a.

Front wheel:

b.

Rear wheel:

3.

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

a.

Front wheel:

b.

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).

6.

a circle is graphed on a coordinate plane with a center point of Z and a radius drawn to the point (5,5). There is a right angle drawn within the circle. xy

7.

a circle is graphed on a coordinate plane with a center point of Z and a radius drawn to the point (5,-5). There is a right angle drawn within the circle. The angle between the x axis and the hypotenuse is undetermined. xy

8.

a circle is graphed on a coordinate plane with a center point of W. A ray is drawn from Z to the point (-4, 4 time the square root of 3) creating a right angle on the x axis and an undetermined angle from the hypotenuse to the x axis. x–6–6–6–4–4–4–2–2–2222444666y–6–6–6–4–4–4–2–2–2222444666000

9.

a circle is graphed on a coordinate plane with a center point of W. A ray is drawn from Z to the point (-4, -4 time the square root of 3) creating a right angle on the x axis and an undetermined angle from the hypotenuse to the x axis. xy

10.

a circle is graphed on a coordinate plane with a center point of W. A ray is drawn from Z to the point (6 time the square root of 3,-6) creating a right angle on the x axis and an undetermined angle from the hypotenuse to the x axis. xy

11.

a circle is graphed on a coordinate plane with a center point of W. A ray is drawn from Z to the point (6 time the square root of 3,6) creating a right angle on the x axis and an undetermined angle from the hypotenuse to the x axis. xy

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 ?

a circle graphed on a coordinate plane with point at (-1,0), (0,1), (1,0), and (0,-1) xy(0, 1)(0, 1)(0, 1)(-1, 0)(-1, 0)(-1, 0)(0, -1)(0, -1)(0, -1)(1, 0)(1, 0)(1, 0)AF

Go

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

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20.