Question

1. A ferris wheel has a radius of 12 m. The center of the ferris wheel is 14 m above the ground. When it is rotating at full speed the ferris wheel takes 10 s to make a full turn. We can track one seat on the ferris wheel. Let’s define t = 0 to be a time when that seat is at the top of the ferris wheel while the ferris wheel is rotating at full speed. (a) Write a function which describes the height of the seat above the ground as a function of time. You should show your reasoning behind constructing this function. At a bare minimum you should probably explain • What are the maximum and minimum heights of the seat? • What is the period of the function? • What is the initial value of the function? (b) The seats on this ferris wheel are separated by an angle of 15◦ around the ferris wheel. Write the functions describing the heights as functions of time for the seats immediately “ahead” and immediately “behind” the seat that you wrote the function for in 2a. (c) The ferris wheel has a frame that is a big circle of radius 12 m. The hinges that support the seats are attached directly to this frame. What is the distance, as measured alone the frame, between hinge supporting one seat and the next one along the ferris wheel? A sketch will probably help you to solve this and will also help you to explain your solution. (d) What is the straight line distance between a hinge supporting a seat and the next one along the ferris wheel? Again, a sketch will be very helpful. 2. In a physics lab, a mass on the end of a spring is being monitored with a motion sensor (basically a small sonar). It oscillates with a height above the motion sensor as a function of time which is described by y(t) = (0.07) sin 2π 0.8 (t − 0.4) + 0.13 where y is in meters and t is in seconds. (As a physicist it makes me cringe to not include units in the equation. But mathematicians don’t like units messing up nice clean mathematical expressions, so I’ll do it their way in this course.) (a) What is the period of this oscillation? (b) Do a detailed sketch of a y vs. t plot of this oscillation. By a detailed sketch I mean that it should have numbers on both axes so that anyone looking at the plot should be able to determine the period, maximum and minimum values of y, and times when y is at a maximum or minimum value. Show at least two full oscillations in the sketch. (c) BONUS: write the function which describes the velocity (technically the y-component of velocity) of the mass as function of time. Show how you got it. 1 MATH 1203 Assignment #2 - Basic Trigonometric Functions Due: Thurs., Jan. 31 3. A person is standing 12 meters away from a streetlight. They observe that they cast a shadow that is 3.5 meters long. If a ray of light from the streetlight to the tip of the persons shadow forms an angle of 27.5◦ with the ground, how tall is the person and how tall is the streetlight? 4. On YouTube find one of the channels “Numberphile” or “Standup Maths”. Look over the videos on the channel you have chosen and pick something that looks like it will contain interesting math. Watch it. (a) Give a brief summary of the video. This should take no more than 2 or 3 sentences. (b) Is the topic of the video connected to what we study in this course? Both “yes”, or “no” are perfectly valid answers, however provide a justification for your opinion.

Answer 1

3. A person is standing 12 meters away from a streetlight. They observe that they cast a shadow that is 3.5 meters long. If a ray of light from the streetlight to the tip of the persons shadow forms an angle of 27.5◦ with the ground, how tall is the person and how tall is the streetlight?

In triangle ECD :

EC = opposite = height of the person = h

CD = adjacent = length of the shadow = 3.5 m

using the equation

tan27.5 = EC/CD

tan27.5 = h/3.5

h = 1.82 m

In triangle ABD

AB = opposite = height of the streetlight = H

BD = adjacent =12+ 3.5 = 15.5 m

using the equation

tan27.5 = AB/BD

tan27.5 = H/15.5

H = 8.07 m