Question

In Buenos Aires, Argentina, the average monthly temperature is highest in January and lowest in July, ranging from 83 F to 57F. Write a cosine function that models the changes in temperature according to the month of the year.
*How can you find the amplitude?
*What part of the problem describes the length of the cycle?

Answer 1

This is a math problem in my math review that i'm having problems with. Any immediate help would be greatly appreciated.

Answer 2

Recall that the amplitude of a cosine curve is the distance from the "zero" to the maximum; in other words, it is equal to half the difference between the biggest value and the smallest value.

In this case, the minimum is 57 and the maximum is 83. So you can find the amplitude of the curve by calculating
\[Amplitude=\frac{ 83-57 }{ 2 }\]

Answer 3

yes so the amplitude is 13

Answer 4

my main problem is trying to figure out what the period is

Answer 5

The temperature is going to vary among an entire year, so the length of an entire cycle has to be 1 year or 12 months. Since the question wants you to graph it in terms of months, the period would have to be 12 months. In other words, the temperature in January will be 57, and then it will go up and down until 12 months later you're back to January and 57.

Answer 6

So would the function model be

F(t)=13cos(pi/6)?

Answer 7

That is what the function would be if the lowest temperature was -13 and the maximum temperature was +13. We have to add one final term to the end to shift the graph up so that the minimum is 57 and the maximum is 83. To do this we want to add 70 to the function so it looks like this:
\[F(t)=13\cos(\pi/6) +70\]
Since cosine goes between -1 and 1. When cosine is equal to 1 your function would only give you 13, but you need to get up to 83 (the actual maximum) so thats why you add the 70

Answer 8

Forgot the x term in there:
\[F(t)=13\cos(x*\pi/6)+70\]

Answer 9

O Thank you alot I was stuck on this forever. I have two other questions that im completely lost on, would you be able to help me with those?

Answer 10

You're welcome, but theres one step left. Remember that cosine would have a maximum at 0, which in this case would be January, and thats not right. We'll want to shift the function, or we can just stick a negative in front
\[F(t)=-13\cos(x*\pi/6)+70\]
Which should be your answer. You can always check an answer like this on your graphing calculator (or one online)

Answer 11

Note that in this model,

January - x=0
February - x=1
.
.
.
December - x=11

Answer 12

Got it thanks alot, could you help me with my other two problems?

Answer 13

Sure, could you post them here on Openstudy in this Math section? That way other users can help as well if I can't

Answer 14

A helix is a three dimensional spiral. The coiled strands of DNA and the edges of twisted crepe paper are examples of helixes. In the diagram, the y-coordinate of each edge illustrates a cosine function. Write an equation for the y-coordinate of one edge.