Sinusoidal applications in AP Pre-Calculus explore real-world scenarios using sine and cosine functions. Topics include modeling the height of a buoy, analyzing Ferris wheel heights, and predicting animal populations. Each problem requires students to derive equations, sketch graphs, and answer specific questions related to sinusoidal behavior. Ideal for AP Pre-Calculus students preparing for exams, this resource provides a comprehensive approach to understanding sinusoidal functions in various contexts.

Key Points

  • Models the height of a buoy in the Chesapeake Bay using sinusoidal functions.
  • Analyzes the height and viewing times of the London Eye Ferris wheel.
  • Predicts fox population changes over time using sinusoidal modeling.
  • Explores tidal patterns in Southern California through sinusoidal equations.
  • Examines solar flare cycles and their periodic nature using sinusoidal functions.
Mayflowers
6 pages
AP® Precalculus
Mayflowers
6 pages
AP® Precalculus
201
/ 6
Pre-Calculus
Name
Sinusoidal
Yhaplications
All
work
must
be
shown
on a
separate
sheet
of
paper.
Write
both
a
sine
and
cosine
function
modeling
each
application,
sketch
a
graph
when
possible
showing
the
sinusoid
and
be
sure
to
answer
all
questions
from
each
application.
1.
A
signal
buoy
in
the
Chesapeake
Bay
bobs
up
and
down
with
the
height
of
its
transmitter
(in
feet)
above
the
sea
level
during
a
small
squall
varying from
1
ft,
to
9
ft.
There
are
3.5
seconds
from
one
9
ft
height
to
the
next
and
when
t=
2
seconds
the
transmitter
is
at
its
highest
point
above
sea
level.
Find
an
equation
that
models
the
height
of
the
transmitter
over
time.
2.
One
of
the
newest
and
largest
Ferris
Wheels
in
the
world
is
the
London
Eye,
a
wheel
that
is
135
meters
high
(more
than
400
feet
high)
and
provides
a
30
minute,
slow
moving
“flight”
over
the
Thames
River,
opposite
“Big
Ben”
in
London,
England.
Suppose
that
we
assume
that
the
London
Eye
has
a
diameter
of
130
meters
and
that
the
lowest
point
of
the
wheel
is
5
meters
above
the
Thames
River.
When
the
London
Eye
is
more
than
110
meters
above
the
Thames,
the
riders
have
a
view
of
Trafalgar
Square.
How
long
will
the
riders
have
this
view
on
their
ride?
3.
Naturalists
find
that
populations
of
some
kinds
of
predatory
animals
vary
periodically
with
time.
Assume
that
the
population
of
foxes
varies
sinusoidally
with
time.
Records
started
being
kept
at
time
t=
0
years,
a
minimum
number
of
200
foxes
appeared
when
f=
2.9
years,
and
the
maximum
of
800
foxes
occurred
at
t=
5.1
years.
Predict
the
fox
population
when
t=
7, 8, 9,
and
10
years.
Foxes
are
declared
an
endangered
species
when
their
population
drops
below
300.
Does
this
ever
happen?
If
so,
when;
if
not,
explain.
4.
On
a
particular
Labor
Day,
the
high
tide
in
southern
California
occurs
at
7:12
AM.
At
that
time
the
water
at
the
end
of
the
Santa
Monica
Pier
is
11
ft.
deep.
At
1:24
PM
a
low
tide
of
7
ft.
occurs.
Assume
the
depth
of
the
water
is
a
sinusoidal
function.
What
time
on
Labor
Day
does
the
first
low
tide
occur?
What
was
the
approximate
depth
of
the
water
at
4:00
AM
and
at
9:00
AM?
What
is
the
first
time
on
that
Labor
Day
that
the
water
is
9
ft
deep?
5.
For
several
hundred
years,
astronomers
have
kept
track
of
the
number
of
solar
flares,
or
“sunspots”,
that
occur
on
the
surface
of
the
Sun.
The
number
of
sunspots
in
a
given
year
varies
periodically,
from
a
minimum
of
about
10
per
year
to
a
maximum
of
about
110
per
year.
Between
1750
and
1948
there
were
exactly
18
complete
cycles.
Assume
that
the
number
of
sunspots
per
year
is
a
sinusoidal
function
of
time
and
that
a
maximum
occurred
in
1948.
How
many
sunspots
will
there
be
in
the
year
2020?
This
year?
What
is
the
first
year
after
2020
in
which
there
will
be
35
sunspots?
What
is
the
first
year
after
2020
in
which
there
will
be
a
maximum
number
of
sunspots?
6.
Assume
that
you
are
aboard
a
research
submarine
doing
submerged
training
exercises
in
the
Pacific
Ocean.
At
time
t=
0,
you
start
porpoising
(going
alternately
deeper
and
shallower).
At
time
¢=
4
minutes,
you
are
at
your
deepest,
1000
meters
below
sea
level.
At
time
¢=
19
minutes,
you
are
at
your
shallowest,
100
meters
below
sea
level.
Assume
that
your
depth
varies
sinusoidally
with
time
and
your
submarine
cannot
communicate
with
ships
on
the
surface
when
the
submarine
is
deeper
than
350
meters
below
sea
level.
When
you
begin
porpoising,
can
the
submarine
communicate
with
ships
on
the
surface?
For
how
many
minutes
is
the
submarine
unable
to
communicate
with
ships
on
the
surface
during
each
porpoising
event?
7.
A
spacecraft
is
in
an
elliptical
orbit
around
the
Earth.
At
time
t=
0,
itis
at
its
apogee
and
its
distance
d=
1000
kilometers
above
the
Earth.
At
time
t=
60
minutes,
it
is
at
its
perigee,
and
its
distance
d=
100
kilometers.
Assuming
that
the
distance
varies
sinusoidally
with
time,
predict
the
first
two
positive
values
of
tfor
which
d=
200
kilometers.
8.
A
theme
park
is
building
a
portion
of
a
roller
coaster
track
in
the
shape
of
a
sinusoid.
You
have
been
hired
to
calculate
the
lengths
of
the
horizontal
and
vertical
support
beams
to
use.
The
high
and
low
points
of
the
track
are
separated
by
50
m.
horizontally
and
30
m.
vertically.
The
low
point
is
3
m.
below
the
ground.
The
vertical
support
beams
are
spaced
2
m.
apart,
starting
at
the
high
point
and
ending
just
before
the
track
goes
below
ground.
The
horizontal
beams
are
spaced
2
m.
apart,
starting
at
ground
level
and
ending
just
below
the
high
point.
The
builder
must
know
how
many
vertical
and
horizontal
beams
to
order.
Find
the
total
length
of
the
vertical
beams
and
the
total
length
of
the
horizontal
beams.
9.
In
New
England
and
other
places,
water
wheels
were
used
for
many
jobs,
such
as
turning
wheels
to
grind
corn.
Typically,
a
stream
flowing
against
the
paddles
on
a
wheel
turn
the
wheel
to
generate
the
power.
Suppose
a
water
wheel
is
2
ft
in
diameter
and
the
lowest
point
on
the
wheel
is
1
foot
under
the
water.
The
wheel
makes
3
complete
revolutions
every
minute
when
the
stream
is
flowing
rapidly.
Suppose
that
you
observe
a
paddle
and
note
that
it
is
at
the
bottom
after
2
seconds.
Predict
the
height
of
the
paddle
at
17
sec.
and
25
sec.
What
are
the
first
3
times
that
the
paddle
enters
the
water
after
you
begin
timing,
and
what
are
the
first
3
times
the
paddle
leave
the
water
after
you
begin
timing?
How
long
is
the
paddle
under
water?
What
fraction
of
the
circumference
of
the
water
wheel
is
under
water
at
any
time,
and
what
is
the
length
of
the
arc
that
is
under
the
water?
10.
Tarzan
is
swinging
back
and
forth
on
his
grapevine,
alternately
going
over
land
and
water.
Consider
values
of
y,
distance
from
the
riverbank,
to
be
positive
if
Tarzan
is
over
water
and
negative
is
he
is
over
land.
Assume
that
Jane
is
measuring
Tarzan’s
motion
and
finds
that
2
seconds
after
she
starts
her
stopwatch,
Tarzan
is
at
one
end
of
his
swing,
where
y=
-23
ft,
When
t=
5
seconds,
he
is
at
the
other
end
of
his
swing
where
y=
17
ft.
Predict
y
when
t=
2.8,
6.3,
and
15
seconds.
Where
is
Tarzan
when
Jane
started
her
stopwatch?
Determine
the
time
indicated
on
the
stopwatch
when
Tarzan
first
reached
a
point
directly
over
the
bank,
11.
Over
the
course
of
a
year,
the
length
of
the
day
(the
number
of
hours
from
sunrise
to
sunset)
changes
each
day.
The
table
below
shows
the
length
of
day
every
30
days’
for
Boston,
Massachusetts
from
December
31,
2007
to
March
26,
2009.
Date
|
12/31
|
1/30
|
3/1
3/31
|
4/30
|
5/30
|
6/29
|
7/29
|
8/28
|
9/27
|
10/27
|
11/26
|
12/26
|
1/25
|
2/24
3/26
|
Day
#
|
0
30
60 90
120
150
180
210
240
270 300
330
360 390
420
450
Hours
|
9.1
9.9,
11.2
12.7
14.0 15.0
15:3
14.6
13.3
11.9
10.6
9.5
9.1
97
11.0
12.4
Rosita
who
lives
in
Boston,
suffers
from
SAD
(seasonal
affective
disorder).
During
the
winter
she
gets
depressed,
but
by
the
first
day
of
spring,
March
21,
she
feels
wonderful.
She
has
been
advised
to
use
light
therapy
during
the
winter,
Assume
that
one
hours
of
light
therapy
replaces
one
hour
of
natural
daylight.
Write
a
model
that
Rosita
could
use
during
the
fall
and
winter
of
2008-2009
(September
21
to
march
21)
to
determine
how
much
light
therapy
she
needs
on
a
particular
day
in
order
to
have
an
equivalent
amount
of
daylight
until
March
21.
How
long
should
Rosita
apply
light
therapy
on
January
15,
2009?
12.
You
seek
a
treasure
that
is
buried
in
the
side
of
a
mountain.
The
mountain
range
has
a
sinusoidal
cross-
section.
The
valley
to
the
left
is
filled
with
water
to
a
depth
of
50
meters,
and
the
top
of
the
range
is
150
meters
above
the
water
level.
The
treasure
is
located
within
the
mountain
at
the
point
(x,
y)
=
(130,
40).
(This
point
is
not
on the
graph!)
Which
would
be
a
shorter
way
to
dig
to
the
treasure,
a
horizontal
tunnel
or
a
vertical
tunnel?
Justify
your
answer.
4y
v7
Mountain
top
13.
As
you
stop
your
car
at
a
traffic
light,
a
pebble
becomes
wedged
between
the
tire
treads.
When
you
start
off,
the
distance
of
the
pebble
from
the
pavement
varies
sinusoidally
with
the
distance
you
have
traveled.
Assume
that
the
diameter
of
the
wheel
is
24
inches.
Predict
the
distance
from
the
pavement
when
you
have
gone
15
inches.
What
are
the
first
two
distances
when
the
pebble
is
11
inches
from
the
pavement?
14.
An
old
rock
formation
is
warped
into
the
shape
of
a
sinusoid.
Over
the
centuries,
the top
has
eroded
away,
leaving
the
ground
with
a
flat
surface
from
which
various
layers
of
rock
are
cropping
out.
Since
you
have
studied
sinusoids,
the
geologists
call
upon
you
to
predict
the
depth
of
a
particular
formation
at
various
points.
A
hole
drilled
at
x
=
100
meters
shows
that
the
top
of
the
formation
is
90
meters
deep
at
that
point.
If
a
hole
were
drilled
to
the
top
of
the
formation
at
x
=
510,
how
deep
would
it
be?
What
is
the
maximum
depth
of
the
top
of
the
formation,
and
what
is
the
value
of
x
where
it
reaches
this
depth?
How
high
above
the
present
ground
level
did
the
formation
go
before
it
eroded
away?
For
what
values
of
x
between
0
and
800
meters
is
the
top
of
the
formation
within
120
meters
of
the
surface?
15. A
phenomenon
is
said
to
be
circadian
if
it
occurs
in
24
hours.
A
person’s
body
temperature
is
circadian,
since
there
are
normally
small,
sinusoidal
variations
in
body
temperature
from
a
low
of
98.2
degrees
F
to
a
high
of
99
degrees
F
throughout
a
24
hour
day.
Use
this
information
to
write
a
circadian
equation
for
a
person’s
body
temperature,
given
t
=
0
corresponds
to
midnight
and-that:a
person
usually
reaches
their
minimum
temperature
at
5
AM;
find
the
time(s)
during
a
day
when
a
person
reaches
“normal”
body
temperature
(98.6)
and
find
the
number
of
hours
each
day
that
body
temperature
is
98.4
or
less.
-«.
16.
The
figure
shows
a
vertical
cross
section
through
a
piece
of
land.
The
y-axis
is
drawn
coming
out
of
the
ground
at
the
fence
bordering
land
owned
by
your
boss,
Earl
Wells.
Earl
owns
the
land
to
the
left
of
the
fence
and
is
interested
in
getting
land
on the
other
side
to
drill
a
new
oil
well.
Geologists
have
found
an
oil-bearing
formation
below
Earl’s
land
that
they
believe
to
be
sinusoidal
in
shape.
At
x
=
-100
feet,
the
top
surface
of
the
formation
is
at
its
deepest,
y
=
-2500
feet.
A
quarter
cycle
closer
to
the
fence,
at
x
=
-65
feet,
the
top
surface
is
only
2000
feet
deep.
The
first
700
feet
of
land
beyond
the
fence
is
inaccessible.
Earl
wants
to
drill
at
the
first
convenient
site
beyond
x
=
700
feet.
Find
the
first
interval
in
the
available
land
for
which
the
top
surface
of
the
formation
is
no
/ 6
End of Document
201

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FAQs

How do you model the height of a buoy using sinusoidal functions?
To model the height of a buoy bobbing in the Chesapeake Bay, you can use a sine or cosine function. The height varies between a minimum and maximum value, which can be determined from the problem statement. The amplitude is half the distance between the maximum and minimum heights, while the period can be calculated based on the time it takes to complete one full cycle. By establishing these parameters, you can create an equation that accurately represents the buoy's height over time.

What is the significance of the London Eye's height in sinusoidal applications?
The London Eye's height is crucial for understanding how sinusoidal functions can model real-world scenarios. With a maximum height of 135 meters and a minimum height of 5 meters, the Ferris wheel's motion can be represented using sine or cosine functions. This allows for the calculation of viewing times at specific heights, such as when riders can see Trafalgar Square. Analyzing these heights helps students grasp the practical applications of sinusoidal functions in engineering and design.

How can sinusoidal functions predict animal population changes?
Sinusoidal functions can effectively model periodic changes in animal populations, such as the population of foxes. By identifying the maximum and minimum population values and the time intervals at which they occur, students can derive a sinusoidal equation. This equation can then be used to predict future population sizes at specific time points, helping to assess conservation efforts and understand ecological dynamics.

What role do sinusoidal functions play in understanding tidal patterns?
Sinusoidal functions are essential for modeling tidal patterns, as they can represent the cyclical nature of high and low tides. By determining the maximum and minimum water depths and the time intervals between tides, students can create equations that predict water levels at any given time. This understanding is vital for coastal management and navigation, as it informs when water levels are safe for docking and other activities.

How do solar flare cycles illustrate sinusoidal behavior?
Solar flare cycles can be modeled using sinusoidal functions, demonstrating the periodic nature of sunspot activity. By analyzing historical data on sunspot counts, students can establish maximum and minimum values, as well as the cycle duration. This modeling helps predict future sunspot occurrences, which is significant for understanding solar activity's impact on Earth and technological systems.