Writing Project 1: (30 Points)
Related Rates and Derivatives in the Real World
We have done a lot of work in the land of Calculus but much of it has been based on abstract concepts and perfecting our calculation abilities. However, Calculus has a rich history of being one of the most directly applicable forms of all mathematics. In fact, the notion of a derivative (rates of change in general as well) were formulated based on solving real world problems related to physics and to the sciences. In today’s day and age, the application of these relations between natural and industrial processes is used to optimize the quality of human life in both professional and personal endeavors. Hence, it will be the central focus of our first writing assignment to look out into the world to see where these amazing ideas can help bring an air of clarity and extend efficiency in some of the most difficult problems that humans may ever face.
Instructions: Write a short (2-3 page) paper in which you answer the following questions:
1.) What is an industry or profession that interests you? Does this profession outwardly use mathematics to solve daily problems? Give a brief introduction to these problems and why they are necessary to solve.
2.) Is there some sort of optimization or rate of change problem these professionals face that can be solved through Calculus? How would these problems be set up algebraically? What are the units of each variable involved?
3.) Set up and solve 2 problems with realistic values pertaining to these aforementioned problems (you do not have to type these but you can). You must take the derivative and compare two different rates of change in at least 1 of these problems. Explain what the solution of each problem means in human language and why the result is important.
Here is a breakdown of how you will be scored on this assignment:
Assignment
Section:
Introduction
Body
Set up of Two Problems
Solve these two problems
Conclusion
Total:
Requirement:
Answer each question posed in 1.) above. You are graded on your thoroughness and your clarity here.
Answer each of the questions posed in 2.). Make sure your equations correctly relate essential information and that both your units and your algebra properly reflects the problem faced.
Write the exact problems with necessary numerical data out in words and then translate that to mathematics.
Show the exact process of solving these two problems. Show all work and explain all steps for full credit here.
Explain what the solutions to your problems mean in English. What conclusions can you draw about your results? Wrap it up nicely and explain what you learned.
Points:
5 Points
5 Points
5 Points
10 Points
5 Points
30 Points
Example: I have written an example problem (DO NOT COPY THIS FOR YOUR OWN WRITTEN WORK). Use this as a way of gathering ideas.
Example:
A Swordsmith needs to heat iron sands up to a temperature of at least 1200 Degrees Celsius for at least 72 hours before the materials can be hammered into a traditional Japanese Katana. Temperature of an object can be modelled basically by Newton’s Law of Heating (there are more involved methods that use more variables but this will do for this assignment’s purpose) which is:
Where is the temperature of the materials, is the temperature of the surroundings, is time in minutes, is the initial difference between the surroundings and the material, and where is a constant depending on the material.
The average temperature of Japan is 29 Degrees Celcius and, thus, the temperature of the iron will initially be around this temperature. Furthermore, forges for this purpose can be assumed to be heated to 1300 Degrees Celsius. Through scientific research it has been found that the constant for iron is roughly (we will assume we have 100 mg of iron here). Hence, we have the equation:
=
A swordsmith might ask the following questions about the heat of these materials:
1.) When is the iron properly heated up to 1200 Degrees?
2.) What is the rate of heat increase when the iron hits 1200 degrees? This might be important to know how far above 1200 degrees the sword is getting and to then adjust the time in the forge.
3.) When is the temperature of the iron increasing at less than 1 degree per minute? This will let the swordsman know when the materials have basically hit their maximum heat level.
Problem 1:
This problem is solved by letting and solving for . This gives : ( )
Thus, the iron is properly heated after minutes.
Problem 2:
First we need to take the derivative. We get:
Thus, at 5.61 minutes we get,
Thus, the temperature of the iron is still heating up at a rate of 45.81 Degrees per Minute when the materials hit 1200 Degrees.
Problem 3:
We will set as the temperature increase should be steadily dropping and thus will increase by less than 1 degree per minute after this point. Hence, ( )
Thus, minutes. So after 14.11 minutes the iron will be increasing at less than 1 degree per minute.
Additional Question:
If the iron melts into a circular pool as it heats where initially the radius of this pool is 1cm and then the radius of this pool increases at a rate of .75cm per 100 degrees increase in temperature, how fast is area of the bottom of this pool increasing when the temperature of the iron is increasing at 50 Degrees per minute?
The area of the bottom circle is . Since the radius of this circle is initially 1 and then increases by .75 for every 100 degrees it increases we have, ( ( ))
Plugging into area of the circle we get,
Since we desire rates of change we will have to take the derivative in terms of time giving us,
Now we need to know what the temperature of the iron sands is when the rate of change of temperature to time is 50. Using a previous equation we have,
( )
Which is roughly 5.42 minutes. Which if we then plug into our initially temperature equation we get,
Hence we have everything we need to solve for change in Area in terms of change in time,
( )
So the area is increasing at a rate of about 22.86 per minute.
