|
Intermediate Algebra Examples
Int Alg Home
| If it is not already on your hard drive, you will need to download
the free DPGraph Viewer to view some of
the pictures linked to on this page. |
QuickTime
6 free download. |
Distance = Rate x Time Examples
Two men leave the same place at the
same time walking in opposite directions. One man (Blue) is walking
at the rate of 2 mph and the other man (Green) is walking at the rate of 3
mph. How long before they will be 6 miles apart? How far apart
are they after 2 hours? Click here
to see an animation. In the animation the elapsed time is shown by
the animated point on the y-axis. The x-axis indicates distance
traveled corresponding to the two animated men. The distance
traveled by Blue is indicated by negative numbers since his direction is
opposite that of Green. Quicktime
animation Solution
(Larger Print)
A red train leaves Altoona heading to
Spoker at 4PM at a speed of 65mph. One hour later a blue train leaves Spoker heading to
Altoona at a speed of 80mph. The distance from Altoona to Spoker is
355 miles. (A) At what time will the trains pass each other?
(B) Which
train will arrive at its destination first? Click
here to see an animation. In the animation the path of the red
train is in red and the time the red train has been traveling is shown vertically in
red. The path of the blue train is in blue and the time the blue
train has been
traveling is shown vertically in blue. Click
here for another version of the animation where little trains are
drawn instead of colored paths. Quicktime
animation Flash video with
audio
Solution
A(Larger Print) Solution
B(Larger Print)
Other Linear Equation
Examples
| A man wants to build a rectangular speaker
box whose volume is 3 cubic feet for his 15 inch (diameter) speakers.
If the square base is 18 inches on each side, how deep are the speaker
boxes? This problem was taken from Intermediate Algebra:
One Step at a Time by Dr. Robert Rapalje.
Rapalje website
Let d = the depth of a speaker box in inches
The volume of the box will be 182
times d in cubic inches. The volume of the box must be 3 cubic
feet which would be 3 times 123 cubic inches.
|
 |
| The center lane of a circular track has a
radius of 100 feet. How many times must you run around the track
to have run a mile? There are 5280 feet in a mile. This
problem was taken from Intermediate Algebra:
One Step at a Time by Dr. Robert Rapalje.
Extra Credit:
Suppose the circular track is located in a
rectangular coordinate system with the center of the circle at the
origin. Find the coordinates of the runner's location after
running exactly one mile if the runner started at the point (100,0) and
ran in a counter clockwise direction as indicated in the animation.
Click here for a hint. |
Click on the track to see an animation of the
blue dot (runner) running the laps needed to run a mile. Quicktime
version |
Applications Involving Quadratic Equations
A
total of 8 meters of fencing are going to be used to fence in a rectangular cage
for pets and divide it into three smaller cages as shown in the animation.
Determine the overall dimensions that will yield a total enclosed
area of 2 square meters. In the animation the total area function is graphed in red.
The blue point moving along the area function corresponds to the changing size
of the rectangular cage. The perimeter of the cage is shown in blue and
the added dividers in green. Let x stand for the length of one of the
sides (with 2 sides needed of length x) and y stand for the length of the other
side (with 4 sections of fencing needed of length y).
| A blue boat is 30 nautical
miles due east of point A and traveling due west at 12 nautical miles per
hour. A green boat is 20 nautical miles due north of point A and
traveling due south at 15 nautical miles per hour. How long until
the distance between the two boats is the square root of 349 nautical
miles? Click
here or on the picture below to see an animation of the next
three hours of the boats' movement. The boats are not drawn to
scale. The endpoints of the red line segment connecting the two
boats represent the location of the boats. The purple graph is
representing the distance between the boats as a function of time.
For extra credit, figure out how long until the
boats are as close together as they will ever get. Quicktime
animation Second Quicktime
animation (This one requires the user to open it using Quicktime
Player but gives more options.)
Consider point A to be at the center of a
rectangular coordinate system. Let x represent the x coordinate of
the blue boat's position and let y represent the y-coordinate of the green
boat's position.
|
 |
 |
| Filling a Paraboloid
The
top picture at the right represents a cross section of
the paraboloid. The paraboloid was formed by revolving the graph of
y = x2, x going from -2 to 2, about the y-axis. Click on
the picture to see an animation.
DPGraph
3D Animation
DPGraph
3D Animation2
DPGraph
3D Animation3
Filling a Paraboloid Using the scrollbar to increase the transparency
in DP animation3 makes for a nice effect.
|
|
| A rectangle is going to be formed
by picking a point in the first quadrant on the graph of the line whose
equation is x + 2y = 6 and drawing line segments perpendicular to the x-axis
and y-axis as shown in the picture. The area of this rectangle will be
xy. Find the values of x and y if the rectangle is to have an area of
4.5 square units.
 |
The line segment that is part of the
graph of x + 2y = 6 is shown in green. Click on the picture to see an animation.
The function that gives the area of the rectangle is graphed in red in the
animation. Quicktime
Version Can you tell which point on the green line segment would
result in the largest (in area) possible rectangle?
The Maths Online
Gallery has a nice applet relating to finding the
rectangle of largest area that can be inscribed in a given right
triangle. After following this link, click on "How to find a
function's extremum".
EC: Give the dimensions of the largest
rectangle that could be constructed by dropping perpendiculars from the
hypotenuse of the right triangle above.
SUPER EC: State a general rule for
constructing the largest such rectangle in any right triangle. |
An Application Involving a Third Degree
Polynomial--Extra Credit
| A rectangular piece of
material measuring 4 ft by 3 ft is to be formed into an open topped box by
cutting equal sized squares out of each corner and folding up the
sides. Determine the size of the squares to be cut out if the
resulting box is to have a volume of 2 cubic feet. There are two
correct answers. Either one will do. One is easier to find than
the other. Super Extra Credit: Determine the size of the squares to be cut out if the
resulting box is to have the maximum possible volume. Click
here to see an animation with scales or on the figure at the right
below to see an animation without scales. Click
here for a 3-D animation without scales and click
here for a 3-D animation with scales. In the animation we see
the changing shape of the material after various sized squares are cut out
of the corners along with the volume function in red shifted to
the right. The 3-D animations show the changing shape of the box
after the sides have been folded up. The animated point moving along the graph of the volume
function corresponds to the changing box construction. Quicktime
animation 3D
Quicktime animation DPGraph
animation Here is a more psychedelic
DPGraph animation. Here are two DPGraph pictures of the box that
would have the largest volume. These can be looked at for entertainment for 15 minutes or so
if you have no life. Picture
of the box one Picture
of the box two (this is picture one with shading) |
x
= the length of a side of each of the squares
|
 |
Applications Involving Radicals
| The formula given below gives the
radius of a sphere as a function of time if the volume of the sphere is increasing at the rate of 8 pi cubic feet per second.
The radius of the sphere will be zero at time t = 0 seconds. Here is an animation showing an expanding hemisphere along with the lengthening
radius. What will be the radius of the sphere 4.5 seconds after time
t = 0 seconds (i.e., at t = 4.5 seconds).
 DPGraphs: expanding
sphere animation
expanding
sphere with radius (activate the scrollbar z-slice feature with the
default z = 0 to see a cross section expanding) |
A ladder
10 feet long is standing straight up against the side of a house.
The base of the ladder is pulled away from the side of the house at the rate of
2 feet per second. How high up the side of the house will the top of the
ladder be 1 second after the base begins being pulled away from the house?
How high up the side of the house will the top of the ladder be after 2 seconds,
3 seconds, 4 seconds, 5 seconds? Here is an
animation of the ladder being pulled away from the side of the house.
Winplot Demonstration (LadderSlide--use T to slide
the ladder) LadderSlide2 (This
one includes a display of the approximate speed of the top of the ladder.)
 |
 Can you see what is happening to the speed
at which the top of the ladder is coming down when the top of the ladder
gets close to the ground? Clicking on the picture produces an
animation. Quicktime Version
Try computing the average speed of the top of the ladder from time t =
4.99999998 seconds to time t = 4.99999999 seconds. This would be
found by computing 
|
Click
here to read about the paradox involved in pulling the bottom of a ladder
out away from a wall at a
constant rate. There is some calculus involved but you might be able to
figure out what the authors are getting at. It's
Extra Credit if you do.
TOP
OF THE PAGE
| 
