Homework 4

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.


Answer the following:

  1. Evaluate the following:
    1. $\tan^{-1}(0)$
    2. $\tan^{-1}(\sqrt{3})$
    3. $\tan^{-1}(1)$
    4. $\sin^{-1}\left(\dfrac{4}{\pi}\cdot \tan^{-1}\left(1\right)\right)$
  2. Find a function that models the simple harmonic motion with the following properties. Assume the displacement is zero at $t = 0$.
    1. Amplitude 4 centimeters, period 3 seconds
    2. Amplitude 8 meters, frequency $\frac{1}{2}$ Hz (cycles per second)
  3. Find a function that models the simple harmonic motion with the following properties. Assume the displacement is at the maximum at $t = 0$ (mass hanging on a spring compressed upwards).
    1. Amplitude 3 inches, period 2 minutes
    2. Amplitude 1.2 meters, frequency $5$ Hz
  4. In a predator/prey model, the predator population is modeled by the function \[y = 1100\cos\left(3t\right) + 500\] where $t$ is measured in years.
    1. What is the maximum population?
    2. Find the length of time between successive periods of maximum population.
  5. A mass suspended from a spring is at rest. Force is introduced at time $t = 0$, causing the mass to complete 15 cycles in 5 seconds. What is the frequency in Hertz?
  6. A mass suspended from a spring is at rest. Force is introduced at time $t = 0$, causing the mass to displace upwwards a maximum of four inches. If the mass completes 4 cycles in one second, find an equation that describes its motion.
  7. A mass suspended from a spring is at rest. It is pulled down three centimeters and released at time $t = 0$. If the mass returns to the location it was released at after one second, find an equation that describes its motion. Hint: It starts at the lowest position.
  8. A mass suspended from a spring is at rest. The mass is compressed upwards 2 centimeters and released at $t = 0$. If the mass first reaches the lowest point under equilibrium at $\frac{1}{3}$ of a second, find an equation that describes its motion.
  9. A trombone player plays the note E and holds the note. Suppose the E causes a variation in air pressure from the normal air pressure with respect to the equation \[V(t) = 0.2 \sin 80\pi t\] where $V(t)$ is measured in pounds per square inch and $t$ is measured in seconds.
    • Find the amplitude, period, and frequency of $V(t)$ with the proper units.
    • If the player increases the loudness of the note, do you think amplitude, period, or frequency will change?
    • If the player plays a lower note instead, do you think amplitude, period, or frequency will change?
  10. For the functions \[y_1 = 10\sin\left(3t - \frac{\pi}{2}\right) \qquad y_2 = 10\sin\left(3t - \frac{5\pi}{2}\right)\] Are they in phase?
  11. Suppose \[y_1 = \cos(2x - 3) \qquad \qquad y_2 = \cos(5x + 5)\] Will these functions ever be in phase? Why or why not?
  12. When you listen to music involving electronic elements, what type of waves are you listening to?
  13. Convert the following to radians:
    1. $150^\circ$
    2. $-60^\circ$
    3. $-390^\circ$
  14. Convert the following to degrees:
    1. $0$ rad
    2. $\frac{\pi}{6}$ rad
    3. $\frac{\pi}{4}$ rad
    4. $\frac{\pi}{3}$ rad
    5. $\frac{\pi}{2}$ rad
    6. $\frac{7\pi}{6}$ rad
  15. Are $-\frac{\pi}{6}$ rad and $330^\circ$ coterminal?
  16. What is the mathematical relationship between the central angle and the arc length?
  17. A circle of radius 9 cm has an arc subtended by an angle of $\frac{5\pi}{6}$ radians. Find the length of the arc and the area of the sector covered by the arc.
  18. Suppose a circle has radius five centimeters. A central angle of measure $40^\circ$ subtends an arc. Find the length of the arc and the area of the sector covered by the arc.
  19. A circle of radius 12 inches has an arc subtended by an angle of 150 degrees. Find the length of the arc and the area of the sector covered by the arc.
  20. Suppose the time is 1:00 PM. How many radians does the minute hand move the clock if it is 1:45 PM? (rotation is clockwise here!)
  21. A central angle $\theta$ in a circle of radius 9 cm is subtended by an arc of length 15$\pi$ cm. Find the measure of $\theta$ in radians and degrees.
  22. A truck with 24 inch radius wheels are rotating at 400 revolutions per minute.
    1. What is the angular speed (in radians)?
    2. What is the linear speed (in inches)?
    3. If instead the truck had 40 inch radius wheels, does the angular speed change? How about linear speed?