Question

In Buenos Aires, Argentina, the average monthly temperature is highest in January and lowest in July, ranging from 83 degrees Fahrenheit to 57 degrees Fahrenheit. Write a cosine function that models the change 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 cycles?

Answer 1

I got the amplitude....it's 13, but I'm not sure about the rest. Is it 70+13 cos ____ because 70 is where the middle of the graph is....

Answer 2

13 is definitely the amplitude and y = 70 is the midline

Answer 3

what is the period or length of each cycle?

Answer 4

erm...not sure in this case....uno momento

Answer 5

\(\pi/6\) because there are 6 months in between January and July, which are the min and max values.

Answer 6

well the period is actually 12 months since things repeat themselves each season (more or less)

so

B = 2pi/T
B = 2pi/12
B = pi/6

so you have the pi/6 correct but pi/6 isn't the period. It's the coefficient for the t value

Answer 7

Oh!

Answer 8

So the function is \[\Large y = 70+13\cos\left(\frac{\pi}{6}t\right)\]
y = average temperature at month t

Answer 9

Ah...I think we missed something...in the back of the book, it says that \(\frac{\pi}{6}\) is the coefficient for (x-1), where x is the month of the year. I have to explain everything I do and it's my last question...any idea how they got that?

Answer 10

the (x-1), instead of just x, is to allow x to start at 1 instead of 0

notice how x-1 = 0 when x = 1

Answer 11

\[\Large y = 70+13\cos\left(\frac{\pi}{6}x\right) ... \text{ x starts at x = 0}\]

\[\Large y = 70+13\cos\left(\frac{\pi}{6}(x-1)\right) ... \text{ x starts at x = 1}\]

x = month number
y = avg temp

Answer 12

Oh! Duh! *face palm*

Answer 13

Again, thank you very much. You're a lot of help lol

Answer 14

you're welcome