**Final Math Project**

Introduction

Periodic Behavior is defined by a “natural systems” which are able to return periodically to the same initial state, going through the same periods and states from which it was once started. (Jick, 2014)

Investigation

Problems

Problem 40 page 453

Solution/Conclusion

We use many oscillating objects in our daily life because they are very common, one example is a rubber band, which behaves very much like a spring, and possesses high elastic potential energy.

One of the most amazing uses of SHM (Simple Harmonic Motion) is music; melodies are composed by the constant movement of oscillating particles, especially cords.

One music device that uses the principle of SHM is the pendulum, which is an object that measures and registers the tempo or speed of music by oscillations. One of the most important discoveries during Renaissance was the pendulum used in clocks

Flowchart

THANK YOU!

References

**Andrea Valeria Macedo**

Montserrat López Orozco

Montserrat Robles Gil Rosas

Monica Liliana González

Montserrat López Orozco

Montserrat Robles Gil Rosas

Monica Liliana González

Circadian rhythms are biological processes that oscillate with a period of approximately 24 hours. That is, a circadian rhythm is an internal daily biological clock. Blood pressure appears to follow such a rhythm. For a certain individual the average resting blood pressure varies from a maximum of 100 mmHg at 2:00 P.M. to a minimum of 80 mmHg at 2:00 A.M. Find a sine function of the form

That models the blood pressure at time t, measured in hours from midnight.

Development of the problem

First of all we had to establish the values of our variables, which are

a

, ,

c

and

b

. The independent value in this function is t because is the time.

So then after we substitute all the values of the function the function becomes this one:

If we had chosen to use degrees, then ω would have been 360 / 24 or 15

a--

is amplitude and is found by (max - min)/2 or (100 mmHg-80 mmHg) / 2 = 10 mmHg

b--

is vertical offset of the sin wave zero points and is found by (max + min) /2 that is (100 + 80) / 2 = 90

--

is a symbol used for frequency and as this cycle is once per day or 24 hours so we use 1/24. We’ll use radians and there are 2π radians in a cycle so the w becomes 2π/24 or π/12

t--

is time

c--

is offset for adjusting the peaks to our base. As our high occurs at 2am or 14 hrs which we want to associate with π/2 or 6π/12 we find c as 6 -14 = -8

Problem 27 page 452

27. Predator Population Model.

In a predator/prey model (see page 432), the predator population is modeled by the function:

y = 900 cos 2t + 8000, where t is measured in years

(a) What is the maximum population?

(b) Find the length of time between successive periods of maximum population.

Mathematical Model

In order to solve this problem we need to know the basic cosine or sine function and its components:

y = a cos (kx)

, where:

A= amplitude

P= 2π/k

So, in order to answer the first question, we need to realize that the amplitude is going to be the maximum population. And we get the amplitude:

y = 900 cos 2t + 8000

a=900

|a|=900

But our formula has a shift, so the maximum population will be more than just the amplitude. We need to add the shift to the amplitude:

|a|=900, shift= 8000

900+8000=8,900

And we got our maximum population that is

=8,900

Graph

In order to solve the second question, we need to realize that the length of time between successive periods of maximum population is equal to the period of the formula.

So in order to get the period, we use the period formula and our formula that is:

y = 900 cos 2t + 8000

if

y = a cos (kx)

=>

k = 2

We can get the period:

P= 2π/k

P= 2π/2

P= 1π

P= π

So

π years

is the length of time between successive periods of maximum population, or every 3,14 years.

An object is in SHM or Simple Harmonic Motion when:

• The acceleration of a certain object is directly proportional to its displacement from its equilibrium position.

• The acceleration is always directed to the equilibrium position. (PhysicsNet)

Simple Harmonic Motion

Example:

Concepts...

Time to play...

PhysicsNet. (2014). Simple Harmonic Motion. Recovered from: http://physicsnet.co.uk/a-level-physics-as-a2/further-mechanics/simple-harmonic-motion-shm/

Jick.(2014).Periodic Behavior. Recovered from: http://www.jick.net/~jess/hr/skept/SHM/node1.html

Science Clarified (2014) Oscillations. Recovered form: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-2/Oscillation-Real-life-applications.html

Stewart, Redlin, Watson. (2009). PRECALCULUS, Mathematics for Calculus. (Fifth Edition). Belmont, CA. Cengage Learning.