Algebra 2 – Day 7 and HW Name____________________________ Date_________ Hour__

Applications of sinusoids – Word Problems

Problem 1: The average daily temperature T (in degrees Fahrenheit) for a certain city is

T = – 23cos2

( 32)365

t

+ 45 where t is the time in days, with t = 1 corresponding to January 1. Find

the average daily temperatures for the following days.

a) January 1

b) October 18 (t = 291 days)

c) Use your graphing calculator to approximate the time of year when the temperature

reaches 44°F.

Problem 2: A company that produces a seasonal product forecasts monthly sales over the next 2 years to be

S = 23.1 + 0.442t + 4.3sin ( )6

t

where S is measured in thousands of units and t is measured in months,

with t = 1 representing January 2012. Predict sales for the following months.

a) February 2012

b) September 2012

c) Use your graphing calculator to approximate when the sales will reach 30 thousands of units for the

first time.

Problem 3: The daily consumption C (in gallons) of diesel fuel on a farm is modeled by

C = 21.6sin2

( 633.2)365

t

+ 30.3 where t is the time in days, with t =1 corresponding to January 1.

a) What is the period of the model? Is it what you expected? Explain.

b) What is the average daily fuel consumption? Which term of the model did you use? Explain.

c) Use your graphing calculator to approximate the times of the year when consumption exceeds

40 gallons per day.

Problem 4: The following chart gives functions which model the wave patterns of different colors of light

emitted from a particular source, where y is the height of the wave in nanometers and t is the length from the

start of the wave in nanometers.

Color Function

Red

Orange

Yellow

Green

Blue

Violet

a) What are the amplitude and period of the

function describing green light waves?

b) The intensity of a light wave corresponds directly

to its amplitude. Which color emitted from the

source is the most intense?

c) The color of light depends on the period of the

wave. Which color has the shortest period? The

longest period?

Problem 5: In a certain forest, the leaf density can be modeled by the equation y = 15 sin 36

( t )

+ 20

where y represents the number of leaves per square foot and t represents the number of months after

January.

a) Determine the period of this function. What does this period represent?

b) What is the maximum leaf density that occurs in this forest and when does this occur? (Sketch a graph to

help you find your answer!)

Problem 6: The population of predators and prey in a closed ecological system tends to vary periodically over

time. In a certain system, the population of snakes S can be represented by 100 20sin( )5

S t

where t is the

number of years since January 1, 2008. In that same system, the population of rats can be represented by

1200 75sin( ( ))

5 2R t

a) What is the maximum snake population and when is it first reached?

b) What is the minimum rat population and when is it first reached?

Problem 7: As Charles swims a 25 meter sprint, the position of his right hand relative to the water surface can

be modeled by the graph below, where h is the height of the hand in inches from the water level and t is the

seconds past the start of the sprint. Write an equation describing the graph.

Problem 8: A buoy oscillates in sinusoidal motion as waves pass in an unknown depth of water. At a given

time it is noted that the buoy moves a total of 7 feet from its low point to its high point, and that it returns to its

high point every 10 seconds. Write an equation that describes the motion of the buoy if, at t = 0, it is at its high

point.

a) Define your variables:

b) Sketch a graph.

c) Write an equation:

Problem 9: The average depth of water at the end of a dock is 6 feet. This varies sinusoidally 2 feet in both

directions with the tide. The high tide occurs at 4 AM. At 10 AM you notice the tide is at its lowest.

a) Define your variables:

b) Sketch the graph.

c) Write the particular equation.

d) Where will the water level be at 3 PM (be careful!)?

Problem 10: The population of dragonflies on a small puddle in the San Antonio River varies sinusoidally with

the days in October. On October 4th the population was at its maximum of 3200 and on October 14th it was at

its minimum of 600 dragonflies.

a) Define your variables:

b) Sketch the graph.

c) Write the particular equation.

d) How many dragonflies where there on Oct. 1st?

e) Using your calculator, find the first day in October when there were 1000 dragonflies.

Problem 11: When a spaceship is fired into orbit it goes into an orbit that takes it alternately north and south of

the equator. Its distance from the equator is approximately a sinusoidal function of time. Ten minutes after it

leaves, it reaches its farthest distance north of the equator, 4000 kilometers. Half a cycle later it reaches its

farthest distance south of the equator, also 4000 kilometers. The spaceship completes an orbit in 90 minutes.

a) Define your variables:

b) Sketch a graph.

c) Write the particular equation.

d) Predict the distance of the spaceship from the equator at 163 minutes.