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Here are a few recommended readings before getting started with this lesson.
During a math lesson, Ali realized many real-life situations that repeat at regular intervals could be modeled by periodic functions. His friends do not believe him. To prove to his friends that trigonometric functions are used in real life, he scoured the internet and found several interesting examples of these functions.
Function rule | What does the function model? |
---|---|
$R(t)=32900sin2t $ | The horizontal distance from a player to a basketball thrown with an initial velocity of $30$ feet per second at an angle with measure $t.$ Negative distance means that the direction of the throw changes. |
$s(t)=3.5sin180π (t+4)+15.7$ | The average wind speed in a certain city, measured in miles per hour. |
$P(t)=100−20cos38π t$ | The blood pressure of a person at rest, measured in millimeters of mercury. |
$N(t)=3.7sin(54t −0.7)+20$ | The number of thousands of employees at a sports company. |
Ali wanted to share the examples with his class, so he printed out the graphs of the functions.
Unfortunately, he forgot to label the graphs. Graph the given functions using a graphing calculator and help Ali match the graphs with their corresponding function rules.
Ali's brother Davontay is a volunteer firefighter. At the end of each year, he likes to analyze certain data related to fire-fighting and rescue services for his department. This year, he received a special assignment from the Fire Chief.
Davontay was asked to collect data about the average monthly water temperature $D,$ in degrees Fahrenheit, as a function of the month of the year $t.$ The table's data then needs to be analyzed. Later, this analysis will test for a relationship between the temperatures and number of rescue boat operations. By analyzing this table, the Fire Chief will let Davontay drive the rescue boat!
Average Monthly Water Temperatures | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
$t$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ |
$D(_{∘}F)$ | $70$ | $73$ | $75$ | $78$ | $82$ | $85$ | $86$ | $85$ | $84$ | $81$ | $75$ | $73$ |
The Fire Chief outlines the mission in more detail, here.
Davontay explains that he and his squad frequently cooperate with medical rescue teams during road accidents. When providing first aid to victims, paramedics measure their vital signs — the electrical activity of a heart in millivolts — using an electrocardiogram (ECG).
A pulse rate of a person is the number of times their heart beats in one minute. Each cycle in the given graph represents one heartbeat.
Write as a fraction
$LHS⋅35 =RHS⋅35 $
$a⋅cb =ca⋅b $
Calculate quotient
Despite analogue radio receivers, SMS, and phone calls, electronic sirens are still used in many fire stations. They create a loud modulated sound that can be heard within a few miles.
Davontay's fire station has a siren that makes $3$ rotations per minute and whose sound beam can be heard within a $4$ mile radius. It so happens that Davontay's house is exactly $4$ miles away from the fire station. Let $D(t)$ be the periodic function that represents the distance from the end of the sound beam to Davontay's house in terms of the time $t,$ measured in seconds.
Number of Rotations | Time |
---|---|
$3$ | $1minute=60seconds$ |
$1$ | $360 =20$ seconds |
Therefore, the siren makes one cycle in $20$ seconds, which is the period of the function.
Although it was sufficient to make an example graph of the distance function, the function rule can also be found. Begin by recalling the coordinates of the points on a unit circle.
These coordinates can be generalized to the coordinates of the points on a circle of different radius. They are known as polar coordinates. Let $r$ be the radius of a circle. The coordinates of a generalized circle are multiplied by $r.$
In the given situation $r$ is equal to $4.$ The angle of rotation depends on the time $t.$ Recall that the period of the function is $20$ seconds. This means that the siren makes a full turn — a rotation of $2π$ radians — in $20$ seconds. Next, the angle $θ_{1}$ by which the siren rotates in $1$ second can be found.Substitute $(4,0)$ & $(4cos10π t,4sin10π t)$
$(a−b)_{2}=a_{2}−2ab+b_{2}$
Subtract term
$(ab)_{m}=a_{m}b_{m}$
Calculate power and product
Commutative Property of Addition
Factor out $16$
$sin_{2}(θ)+cos_{2}(θ)=1$
Identity Property of Multiplication
Add terms
Split into factors
Factor out $16$
$a⋅b =a ⋅b $
Davontay originally volunteered as a firefighter because they live near a volcano along the coast of a seismic zone — an area where earthquakes usually occur. A few years ago, on a dark cloudy day there was danger of a looming earthquake. Davontay's fire department had no choice but to organize the evacuation of all the town's homes.
Davontay was told that the predicted earthquake wave had a frequency of $61 $ hertz and an amplitude — displacement of the ground — of $5$ millimeters.
$Frequency=61 $
$b/ca =ba⋅c $
Identity Property of Multiplication
$1a =a$
$Period=6$
$LHS⋅∣b∣=RHS⋅∣b∣$
$LHS/6=RHS/6$
$ba =b/2a/2 $
$b≥0:b=3π b<0:b=-3π (I)(II) $
Finally, by connecting the points with a smooth curve and continuing it periodically in both directions, the sinusoid can be drawn.
$t=28$
$ca ⋅b=ca⋅b $
Write as a sum
Write as a sum of fractions
Calculate quotient
Split into factors
$sin(θ)=sin(θ+n⋅2π)$
$a(-b)=-a⋅b$
$a⋅cb =ca⋅b $
On the weekends, Ali likes to help Davontay do some cleaning at the fire station. This particular time, Davontay asks his brother to download a mobile app to tune a guitar left at the station by a firefighter who plays it during some downtime. Not everything at a station is related to an emergency!
The standard tuning of a guitar is A–$440,$ which means that the same note A on a keyboard will vibrate $440$ times per second. After downloading the app, the guitar player showed up and excitedly came over to talk about sound waves with Ali.
$LHS⋅∣b∣=RHS⋅∣b∣$
$b1 ⋅a=ba $
$LHS⋅440=RHS⋅440$
$b≥0:b=880πb<0:b=-880π (I)(II) $
It is given that the amplitude is $60$ decibels. In Part A, it was also obtained that the period is $4401 $ seconds. Next, since the cosine function is not translated, the midline of this function is $y=0.$ Graph this line on a coordinate plane.
Now, key points ranging over one cycle will be plotted. These points are the maximums, minimums, and intersections with the midline. The maximums of $y=cosx$ occur once every cycle at the following $x-$coordinates.The minimum of the function is located halfway between the maximums at $x=8801 .$ The minimum value is $0−60=-60.$ Plot the minimum in the same coordinate plane.
Next, between every neighboring maximum and minimum the sinusoid intersects with the midline.Finally, by connecting the points with a smooth curve, the sinusoid can be drawn. Although the graph may not represent the whole sound wave, keep in mind that only one cycle of each graph is given in the options.
This graph corresponds to option B.
The amplitude of a periodic function is half the difference between the maximum and minimum values, while the period is the length of one cycle.
The amplitude is related to neither period nor frequency. Therefore, when the frequency decreases $2$ times, the amplitude does not change. This means that the pitch of a sound does not influence its volume, which makes perfect sense. The guitar player is thankful for this understanding.
One of the most important and valuable fire brigade operations is qualified first aid. Ali admires his brother for conducting first aid training for people that do not have professional knowledge or experience in this field. The basic emergency procedure taught during training is cardiopulmonary resuscitation (CPR) — chest compressions often combined with artificial breaths.
Trainees practice CPR on a medical-training mannequin. CPR for an adult person consists of cycles of $30$ chest compressions and two breaths. Compressions should be $5$ to $6$ centimeters into the mannequin's or person's chest at a rate of $100$ to $120$ compressions per minute.
$a−(-b)=a+b$
Identity Property of Addition
$∣3∣=3$
$a≥0:a=3a<0:a=-3 (I)(II) $