I don't know how to do 182:

Question

180.A large circular saw blade with a 1-foot radius is mounted so that exactly half of it shows above the table. It is spinning slowly, at one degree per second counterclockwise. One tooth of the blade has been painted red. This tooth is initially 0 feet above the table, and rising. What is the height after 37 seconds? After 237 seconds? After t seconds? Draw by hand a graph that shows how the height h of the red tooth is determined by the elapsed time t. It is customary to say that h is a function of t.

181.(Continuation) Now explore the position of the red saw tooth in reference to an imaginary vertical axis of symmetry of the circular blade. The red tooth is initially one foot to the right of the dotted line. How far to the right of this axis is the tooth after 37 seconds? After 237 seconds? After t seconds? Draw by hand a graph that shows how the displacement p of the red tooth with respect to the vertical axis is a function of the elapsed time t.

182.(Continuation) The graphs of the height h and the horizontal displacement p of the red saw tooth are examples of sine and cosine curves, respectively. Draw y = sinx and y = cosx on a graphing tool and compare them with the graphs that you drew in the preceding exercises. Use these graphs to answer the following questions:

(a)For what values of t is the red tooth 0.8 feet above the table? 0.8 feet below the table? (b)When is the tooth 6 inches to the right of the vertical axis? When is it farthest left?

182 part b is what I don't know how to do. What does it mean, 6 inches from the right of the vertical axis? What does it mean, farthest left? The first two questions are quite easy.

Answer 1

Hi Eric (great name by the way),

Just to verify your answers for 180 and 181

180:

sin(37) = 0.60 ft.

sin(237) = -0.84 ft.

h(t) = sin(t)

181:

cos(37) = 0.80

cos(237) = -0.54

d(t) = cos(t)

182) This whole question relies on familiarity with your graphing calculator. I'l assume you have a TI-83 or something similar.

First, click the "Mode" button and change from Radians to Degrees

Then, go to "Window" in your graph settings and set the following parameters:

xmin = 0

xmax = 360

xscal = 90

ymin = -1

ymax = 1

yscal = 1

For part a, you want to know when the tooth is 0.8 feet above the red table. Since you're interested in a vertical displacement, you'll use sin(t). Punch in the following functions:

y1 = sin(x)

y2 = 0.8

Plot both of them.

You'll see a horizontal line intersect the wave in two locations near the left side. Go to the "Calc" function (it should be above the Trace button. Hit "2nd" + "Trace" to go to the menu). You'll have a function called "Intersect". Select that feature.

You'll be returned to the graph. It'll ask you to select the first curve (most likely your sine wave). Click "Enter". Then it'll ask to click your second curve (most likely the line). Click enter. It'll then ask you to "Guess". Use your arrows to drag the cursor near the first intersection point on the left. The calculator will pinpoint the x-value and report it at the bottom. It should say "x = 53.13".

Do the same procedure for the second intersection point. It should say "x = 126.87"

Thus, the tooth is above 0.8 feet between 53.13 seconds and 126.87 seconds.

For part b), 6 inches = 0.5 feet. You're interested in horizontal displacement now, so you'll use cos(t) instead of sin(t).

Change your functions to the following:

y1 = cos(x)

y2 = 0.5

Do the same procedure with intersection points described above. You should get one intersection very close to the left side of the window, and the other intersection very far away on the right side of the window.

The left value should be x = 60. The right value should be x = 300.

Therefore, the tooth is 6 inches to the right of the imaginary line from 0 to 60 seconds, and from 300 to 360 seconds.

Hope that helps!