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Question 1171514: 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. 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 a graph on the grid below 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.
2. 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 a graph on the grid below that shows how the displacement p of the red tooth with respect to the vertical axis is a function of the elapsed time t.
(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. Graph the equations y=sinx and y = cos x on your calculator, and compare these graphs 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?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break this down into the requested parts.
**1. Height (h) as a Function of Time (t)**
* **Understanding the Motion:**
* The blade rotates 1 degree per second.
* The radius is 1 foot.
* The initial height is 0 feet.
* The maximum height is 1 foot.
* **Calculations:**
* After 37 seconds, the tooth has rotated 37 degrees.
* After 237 seconds, the tooth has rotated 237 degrees.
* After t seconds, the tooth has rotated t degrees.
* **Height Formula:**
* The height is determined by the sine of the angle: h(t) = 1 * sin(t degrees)
* To convert degrees to radians for many calculators, the formula would be h(t) = 1 * sin(t*pi/180)
* h(37) = sin(37°) ≈ 0.6018 feet
* h(237) = sin(237°) ≈ -0.8387 feet
* h(t) = sin(t°) feet
* **Graph:**
* The graph will be a sine wave oscillating between -1 and 1.
* The x-axis will be time (t).
* The y-axis will be height (h).
* The graph starts at (0,0) and rises.
**2. Horizontal Displacement (p) as a Function of Time (t)**
* **Understanding the Motion:**
* The displacement is measured from the vertical axis of symmetry.
* The initial displacement is 1 foot to the right.
* The displacement is determined by the cosine of the angle.
* **Calculations:**
* p(37) = cos(37°) ≈ 0.7986 feet
* p(237) = cos(237°) ≈ -0.5446 feet
* p(t) = cos(t°) feet
* **Graph:**
* The graph will be a cosine wave oscillating between -1 and 1.
* The x-axis will be time (t).
* The y-axis will be displacement (p).
* The graph starts at (0,1)
**3. Comparison with Sine and Cosine Curves (y = sin x and y = cos x)**
* **Calculator Graphs:**
* Graph y = sin(x) and y = cos(x) on a graphing calculator.
* Note the similarities to the h(t) and p(t) graphs.
* The main difference is the units of the x-axis (radians vs. degrees) and the speed of the oscillation.
* **(a) Height 0.8 feet:**
* Solve sin(t°) = 0.8 and sin(t°) = -0.8.
* Using a calculator, sin⁻¹(0.8) ≈ 53.13° and 180°-53.13° ≈ 126.87°. Because the sin function is periodic, add 360 to each of those values, and repeat.
* sin⁻¹(-0.8) ≈ -53.13° which is the same as 306.87°, and 180°+53.13° ≈ 233.13°. Add 360 to each of those values, and repeat.
* t ≈ 53.13 + 360n, 126.87 + 360n, 233.13 + 360n, 306.87 +360n, for integer n.
* **(b) Displacement 6 inches (0.5 feet) to the right:**
* Solve cos(t°) = 0.5.
* cos⁻¹(0.5) = 60°. Because the cosine function is periodic and symmetrical about the x axis, 360-60 = 300. So 60 degrees, and 300 degrees are solutions. Add 360 to each of those values, and repeat.
* t ≈ 60 + 360n, 300 + 360n, for integer n.
* **(c) Farthest left:**
* The farthest left is when cos(t°) = -1.
* This occurs at 180°.
* t ≈ 180 + 360n, for integer n.
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