# How Do You Model Desert Temperature Variations Mathematically?

• Quantum_Grid
In summary, the conversation discusses finding a possible formula for the desert temperature, H, in terms of time, t, measured in hours from 5 am. The formula is determined to be H=60+(-20)cos((pi/12)t) and is confirmed to be correct. The conversation also mentions the importance of considering the amplitude and period when creating an oscillation formula.
Quantum_Grid

## Homework Statement

The desert temperature, H, oscillates daily between 40 degrees F at 5 am, and 80 degrees F at 5 pm. Write a possible formula for H in terms of t, measured in hours from 5 am.

## The Attempt at a Solution

The best I can come up with is

H=60+40sin((pi/6)t)

but this does not look right when I try and graph it on a calculator. I think the 40sin part is what I have wrong, but I cannot figure out what goes there, and the book is no help at all. Am I at least close?

The amplitude of the oscillation is 40, but multiplying the base function by 40 will make it oscillate between -40 and 40, giving it an amplitude of 80. Ie., your function oscillates between 60 - 40 = 20 and 60 + 40 = 100.

OK, I looked a little closer and I think they want it to start at 40. So I got H=60+(-20)cos((pi/12)t)

I think I had the period wrong too... This looks right...right?

Quantum_Grid said:
OK, I looked a little closer and I think they want it to start at 40. So I got H=60+(-20)cos((pi/12)t)

I think I had the period wrong too... This looks right...right?

Looks great!

## What is a periodic function?

A periodic function is a mathematical function that repeats its values at regular intervals or periods. This means that the function will have the same values at certain points, and will then repeat those values over and over.

## What is the formula for a periodic function?

The formula for a periodic function is f(x) = A*sin(Bx + C) + D, where A is the amplitude, B is the period, C is the phase shift, and D is the vertical shift.

## How do you determine the amplitude of a periodic function?

The amplitude of a periodic function is the distance from the midline of the graph to the highest or lowest point. It can be determined by taking the absolute value of A in the formula f(x) = A*sin(Bx + C) + D.

## What is the period of a periodic function?

The period of a periodic function is the length of one complete cycle of the function. It can be determined by finding the value of B in the formula f(x) = A*sin(Bx + C) + D and taking the reciprocal of that value.

## How do you find the phase shift of a periodic function?

The phase shift of a periodic function is the horizontal shift of the graph. It can be determined by finding the value of C in the formula f(x) = A*sin(Bx + C) + D and taking the opposite of that value.

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