Wednesday, March 31, 2010

Today's Slides: March 31, 2010

Hello All,
Here are the slides from today's class on the rules of integration.

Cheers, Bru

Tuesday, March 30, 2010

Today's Slides: March 30, 2010

Hello All,
Here are the slides from Exploration 5.1 and our introduction to integrals and integration.

Cheers, Bru

Friday, March 5, 2010

Math Final Group 2

Math Final Project Group 2
Group Member:
YDplusSB, Hyunhwa, J-tron foteen
Materials:

A measuring cylinder
A right circular cone
A plastic bag
A glass
A timer
3 people….

Purpose:

The purpose of our final project is to help us understand calculus espeically the related rate problems better by applying the math into a real world situation. Throughout the process of solving the problem, we could also learn how to edit a video, how to distribute work, and how to work together as a group.

Scenario:
There is a right circular cone with a height of 11.3cm, a radius of 4cm. Initially, there is 15ml amount of water in the cone with a height of 4.85cm. As we unlock the bottom of the cone, the water in the cone will drip with a flow rate of 1.875ml/sec. At the same time, some water is poured into the cone at a flow rate of 0.275ml/sec.

Question:
What’s the rate of change of the radius after 3sec with the volume of 10.45cm3 and the height of 4.226cm?

Little Video:


The actually video quality is low. We use a white backgroup to highlight the actual experiment which consists of a right circular cone, a plastic bag, one glass, and a measuring cylinder.

Check out our solution in WIKI

http://crmscalc2010.pbworks.com/Final-Project#

Ski Movie Final Project

The purpose of this final project is to help us understand related rates through teaching them ourselves. We are also hoping that we can help to teach others and challenge them to a new Related Rates problem. About all everyday scenarios, such as skiing, contain some bit of calculus. We are proposing a scenario, in which two skiers are skiing away from each other. We are going to calculate the rate at which the distance between them is changing. This problem was created by skirdude, secret and Flying Slug.

Two skiers are having an extreme day on Snowmass Mountain. They have been skiing together all day, dropping fatty cliffs and schralping mad gnar. Its the end of the day, and they are getting tired, so they now decide they both want to go down different runs. They are total math buffs and decide to turn this into an awesome Related Rates problem!


You can find our solution here!

And a big shout out to the crew at Aspen for being very patient with us and giving us tons of help!

Thursday, March 4, 2010

Real World Related Rates

Final Project Group 1
Related Rates
Co-Authors: Blitzen, Dammitimmad, and mc Casper.

Materials:
Bowling Ball

Marble Video
Camera
Smooth floor (with minimal friction)
Paper and pen to do the work
Black Box (calculator)
Grey Box (brain)

Purpose: For our final project we were asked to create a related rates problem within the real world using a program to draw the actual data from our experiment. Using the video analysis with a program called logger pro we had all of the data needed to solve the problem at hand.

The Problem: Our experiment consisted of rolling a b
owling ball and a marble away from each other at a 90 degree angle. We executed this experiment on a smooth and level concrete floor indoors. We can assume that both balls were rolling at approximately the same speed (see next paragraph below). Our goal is to find the rate of change of that the distance between the two balls is increasing with respect to time.

The great lengths of assuring a constant velocity:
Time frame of experiment was roughly 2 seconds
Very smooth concrete floor used in experiment
No interference after the initial push off
Air resistance is negligible due to being indoors and
no wind

Figure 1:
Video of actual experiment!
video


Data:

Graph of our data:
You can see from our graph above that the velocity of the two objects is within minor human error, constant. The only part of the lines that are not constant is the beginning portion of the graph due to the ball's being stationary at the beginning of the video. The slight curve that is also apparent before the constant velocity is reached is due to the initial push from us accelerating the balls and bringing them up to speed.

Using the linear regression on our program (called logger pro), we got the equations for the velocity and displacement of each object individually.

Graphs with the linear regression equations.
(Because the linear equations are difficult to read, you can go to our WikiWork Problem to see the equations of these lines more clearly.)
http://crmscalc2010.pbworks.com/WikiWork-Problem-2

General Overview How To Solve:
After obtaining the data and the linear regressions in loggerpro, we found the velocities using the data begotten from loggerpro. We used the X and Y components of object's velocity to find their total velocity.
Next we chose a random point somewhere from the middle of the graph.
To find the rate of change of the distance between the two objects we used the derivative of Pythagorean formula (see wiki). We substituted in the displacements for both objects into the formula and their velocities to find dz/dt (the related rate of the distance between the two objects with respect to time).

We choose to repeat this process for three separate points to see how accurate and consistent our answers were.

After completeing our project we determined that there was very little human error in our experiment because all three final answers were only one tenth different from one another.
***See WikiWork problem for full explination of calculations.***

Final Problem (one last time):
What is the rate at which the distance between the two objects is increasing? The velocity of the bowling ball is 37.957 inches per second. The velocity of the marble is 43.52 inches per second.

***Please see our WikiWork page to see the calculations and solution to this problem***
http://crmscalc2010.pbworks.com/WikiWork-Problem-2

Final Project: The Related Rates Time Machine

Group 4
Nom de plumes: babar, WinnPlot, Tubby.

The purpose of this Final Project was to create a unique related rates problem from a real world situation and then solve this problem, thus expanding our knowledge of related rates, and their application in the real world.




image source:http://www.friedpost.com/wp-content/uploads/2008/10/time-machine.jpg


Young Freddy Newton was trying to make it big in the stock market, and decided to build a time machine (see image) so he would be sure to buy the right Stock. Unfortunately the time machine malfunctioned sending Freddy into a ripple of time and space and landing him in the land of dinosaurs with only a bendy straw in his possession. Freddy was so thirsty upon arrival but could not find any clean water. Finally after three weeks of wandering the land aimlessly Freddy came upon a bowl, that we assume is a half-sphere filled with prehistoric sludge. It had a volume of 355cubic centimeters and a radius at the top of 6.75centimeters. It takes him 22.65 seconds to consume the sludge. Freddy uses his handy dandy bendy straw to drink the sludge for sustenance. We are going to assume that he drinks at a constant rate. Below is a video of this event:




So, the related rates question is:
If a semi-sphere with a volume of 355 cubic centimeters filled with prehistoric sludge takes 22.64 seconds to consume what is the rate, in minutes, at which the height of sludge in the bowl is changing at the moment when the sludge is at 2 centimeters?


Here is a hyperlink to the solution:
http://crmscalc2010.pbworks.com/Final-Project-Group-4#view=page

Wednesday, March 3, 2010

Group 5's Final Project Presentation

MMMhmmm....Ice Cream....


Beston, Marley, and BlueElephants's Final Project Presentation


The purpose of this final project is for us to watch, learn, and teach a real world application of calculus. Our problem involves a conical container and a volume of liquid that is flowing out of the bottom. The purpose is to find what the rate of change of the volume with respect to time is as the liquid flows out at a certain height.
In our problem we measured the rate of change of the ice cream flowing out of an ice cream cone with respect to time.

RELATED RATES
PROBLEM:
An ice cream cone has a radius of 2.5 cm at the top and has a height of 10.2 cm. If the height of the melted ice cream is decreasing at a rate of 0.35228 cm/s, how fast is the volume of the melted ice cream decreasing when the height is 8.271cm?

Materials:

-ice cream cone
-ice cream
-microwave
-knife

Instructions:

Take the ice cream cone and cut off the very tip so that there is a small hole. Measure radius and height of cone. Melt ice cream in microwave. Fill the ice cream cone up with melted ice cream. Film using loggerpro. Move knife down as level of ice cream decreases. Using loggerpro, plot data points based off of level of knife in relation to the base of the cone.




Here is a delightful video of us, showing you, our data collection...





To see our solution please go to the Wiki.


MMMhmmm....Ice cream...But don't let it all melt away...



Photo Credits:

Lady eating ice cream:http://repossibility.com/wp-content/uploads/2009/08/Melting-Ice-Cream-Lady-300x300.jpg
Earth melting:http://collegian.csufresno.edu/a/wp-content/uploads/2008/02/f_icecream.jpg