Problems with linear-to-linear functions
Spr. 05 - 1. Bo is growing his hair out. Today,
his hair is 30 cm long. One year from today, it
will be 35 cm long. Five years from today, it
will be 40 cm long. If his hair’s length is a
linear-to-linear function of time, how many
years from today will it be before his hair is 41
cm long?
------------------------------------------------------------Fall 04 - 1. Shirley knows that the more weight
she loads on her bicycle, the slower she will
have to ride up Morgan’s Hill. If she carries no
load, she can ride 12 feet per second up the
hill. With a load of 50 pounds, her speed drops
to 6 feet per second. With a load of 100
pounds, her speed is 5 feet per second.
Assuming her speed is a linear-to-linear
function of her load, what will her speed be at
a load of 25 pounds?
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Problems involving Locations on a Circle
Winter 03 - 1. (8 points) In the figure, the circle
has radius 10 inches. Calculate the
coordinates of the points A and B.
------------------------------------------------------------Spr. 05 - 4. Angel and Bernard start running
around a circular track. The track has radius
50 meters. They start at the same point, but
run in opposite directions. Angel runs at 4
meters per second, and Bernard runs at 5
meters per second. After running for 10
minutes, how far are they (in a straight line)
from each other?
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Locations on a Circle Continued
Fall 04 - 5. Susanne took a ride on a ferris
wheel which had a radius of 22 meters. She
began her ride at the lowest point on the
wheel, just 4 meters above the ground. During
her ride, her linear speed was 14 meters per
second. How high above the ground was she
86 seconds into her ride?
---------------------------------------------------------Spr. 03 - 3. (14 points) Your seat on a ferris
wheel is at the indicated position at time t = 0.
The wheel has a radius of 30 feet and is
rotating counter-clockwise at a rate of 10
RPM. The bottom of the wheel is 5 feet off the
ground. You find it takes 4 seconds to reach
the bottom of the ride for the first time. Impose
a coordinate system by clearly marking your
choice for the origin in the picture above. Give
the x- and y- coordinates of your position t
seconds after the start of the ride.
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Circular Motion (Belt and Wheel)
Spr. 03 - 2 (6 points) Two wheels are
connected by a belt. The larger wheel has
radius 16 inches and rotates at a rate of 50
RPM. The smaller wheel rotates at a rate of
112 RPM. Compute the radius of the smaller
wheel.
-----------------------------------------------------------Fall 03 - 1. Four pulleys (A, B, C, and D) are
attached by two belts as shown in the figure.
Pulleys A and B are rigidly attached to the
same axle. The pulleys have radii as follows: A
0.85 cm, B 1.75 cm, C 0.4 cm, and D 0.75 cm.
Pulley A has an angular speed of 5 revolutions
per minute. (a) (5 points) What is the angular
speed of pulley C?
(b) (5 points) What is the linear speed of a
point on the belt connecting pulley B and
pulley D?
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Circular Motion Continued
Winter 03 - 2. (10 points) You are riding a
bicycle along a level road. The front sprocket
has radius 5 inches and the rear sprocket has
radius 2.5 inches. You pedal, turning the front
sprocket 85 RPM, as the bicycle travels at a
rate of 26.4 feet per second. What is the
radius of the rear wheel in inches?
-------------------------------------------------------------Winter 04 - 3. Paul is riding a ferris wheel with
a diameter of 250 feet. It is powered by a 1.2
foot diameter motor wheel, which is attached
by a chain to a 26 foot diameter drive wheel,
as shown in the diagram. The drive wheel is
attached to the same axle as the ferris wheel.
The axle of the ferris wheel is 140 feet above
the ground. How fast is the motor wheel
turning if Paul is moving 25 feet per second?
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Problems involving Trig and Angles
Spr. 05 - 3. Godzilla is approaching at 5 feet
per second. From the ground, you measure
the angle to the top of Godzilla’s head to be 17
degrees (that is, the angle between Godzilla’s
head, you, and the horizon is 17 degrees).
One minute later, you measure the angle
again, and find it to be 23 degrees.
Assuming that Godzilla’s height is constant,
how tall is Godzilla?
-------------------------------------------------------------Winter 04 - 5. You are watching a rocket
launch. A short time after take-off, the rocket
appears to be 68 high (i.e., a line from you to
the rocket makes a 68 angle with the
horizontal). A little later, the rocket has climbed
an additional 100 meters, and now appears to
be 70 high. How far are you from the launch
pad?
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Trig and Angles continued
Fall 04 - 4. Find the coordinates of the point P
in the figure below.
-------------------------------------------------------------Spr. 03 - 2. (4 points) If sin  = 4/17 , find the
two possible values of cos . (Either give an
exact answer or round your final answer to
four digits after the decimal.)
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Fall 03 - 4. From your viewpoint, a vertically
rising plume of smoke makes an angle of 54
with the horizontal. You decide to put more
distance between yourself and the plume, and
move 100 meters farther away. You measure
the angle of view again and find it to be 48_,
but the plume has grown 20 meters taller in
the time between your measurements. How
tall is the plume?
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Problems involving inverses
Fall 04 - 2. Let f(x) be defined by
3x  5
f(x) =
x6
Find f-1(x).
-------------------------------------------------------------Winter 04 - 1. Let f(x) = x  x  2 . Find f-1(x).
-------------------------------------------------------------Winter 02 - 2. (8 points) Let y = f(x) = (x + 3)2 +
5 on the domain 3 < x < 7.
(a) What is the range of y = f(x)?
(b) Find a formula for f-1(x).
(c) Sketch a graph of f-1(x), clearly indicating
the domain and range.
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Problems involving multipart functions
Fall 02 - 3. Give the multi-part rule for the
following function:
g(x) = |x| + |x + 3|
-----------------------------------------------------------Winter 03 - 3. (8 points) Let
f(x) =
g(x) =
Give the multipart rule for the function
h(x) = f(x) + g(x).
------------------------------------------------------------2. (9 points) Recall the basic step function
u(t) =
Find the multi-part formula for
h(t) = 5u(t)-u(t-3).
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