Section+1.13

Wyatt Bensken, 5th Period, Wednesday September 14.

We started class today by discussing the "fast" way to do #13 in section 1.12 (see book). This conclusion reached by this discussion was seeing that the Period=7, and that in parts a-d, each one was simply the last plus a period. We also noted that it would be easiest to only calculate "a" because by using 4 it creates a zero which makes it much easier!

Following this discussion we looked at **Section 1.12 #10**. In that problem we were given a graph that looked like this (Wolfram Alpha LLC. Wolfram|Alpha. http://www.wolframalpha.com/input/?i=g%28x%29%3Dcos%28x%29sin%28x%29 (access September 15, 2011)):

We are told that the equation is g(x)=cos(x)sin(x), and that we should assume it is sinusoidal. We are then asked to find the following:

1) Amplitude: It was clear from the graph that the amplitude was equal to- math $\frac{1}{2}$ math

Reminder Amplitude is the distance from the axis of oscillation to the maximum or the minimum.

2) Vertical Displacement: None! We concluded that there is no vertical displacement because the x-axis is halfway between the maximum and minimum values.

3) Period: The period was evident from the graph as well, and that was decided to be- math $\pi$ math

Next Ms. Tyson asked the question why was there no phase shift to this problem? The class stated that there is no phase shift because the middle value hits the y-axis therefore it can be written as sine without the use of a phase shift. {Mrs. Tyson adds you need more infor than just this, can someone give the additional info to the author}

After we discussed this Ms. Tyson asked the class to re-write this using the Period, and Vertical Displacement.

This lead to the following:

math $\sin(x)cos(x)=\frac{1}{2}sin2x$ math

We then changed it to:

math $\ 2sin(x)cos(x)=sin2x$ math

After completing this problem we looked at **Section 1.13 CYU 1-3**.

This is the Ferris wheel problem where we are told that it has a radius of 125 feet, and a maximum height of 264 feet. We were also told that it made a full rotation every 9 minutes (this is the period!). "t" was also defined as the time in minutes after being at the __peak__ Given this information we were tasked with creating a graph, which we did after we wrote a formula.

We worked through it like this:

Period: It was told to us that it was 9 Amplitude: We were told that the radius is 125, which is also the amplitude! Axis of Oscillation: This was a bit tricky because we knew the maximum was 264, however we needed to find the minimum. We found this by doubling the radius (giving you the diameter) then subtracting this from the maximum. Doing this we realized that it sat 14ft about the ground. Using this information we found an Axis of Oscillation at y=139.

Reminder that A of O is =

math $\frac{max+min}{2}$ math

Using this information we wrote the following equation for H(t), the Height at time "t".

math $h(t)=125cos(\frac{2\pi}{9}t)+139$ math

We then proceded to graph it and the result was something like this (Wolfram Alpha LLC. Wolfram|Alpha. http://www.wolframalpha.com/input/?i=h%28t%29%3D125cos%28\frac{2\pi}{9}t%29%2B139 (access September 15, 2011)):

It needs to also be noted that the reason cosine was used (even though you don't get on at the top of the ferris wheel) was because t was defined as the time in minutes after being at the __peak__.

It can also be written as a sine function, by simpling adding a phase shift in of 2.25 (think that to get the middle value you want to shift it .25 of the period and .25 of 9= 2.25). The resulting equation is the following:

math $h(t)=125sin(\frac{2\pi}{9}(t+2.25))+139$ math

At the end of class we also briefly began to discuss the use of negative cosine in this problem as well, although due to lack of time we didn't fully flesh out the idea.