The Great Applied Problem Introduction and Brief History

THE GREAT APPLIED PROBLEM – INTRODUCTION AND BRIEF HISTORY

This problem came to me via a person living in the community in which I teach. A gentleman had a cylindrical tank that was lying horizontally on the ground. Its diameter was 14 feet and its length was 20 feet. The depth of the water in the tank was 4 feet. He wanted to know:

a) How many gallons of water were in the tank?

b) How many more gallons of water will it take to fill the tank?

At first I thought that this was a fairly trivial problem and that I would have his answers in a few minutes. However when I started to reason it out, it became apparent that the solution was much more involved. After I completed the solution, I realized that this problem had more mathematics interwoven in its solution than any other mathematics problem I have ever encountered. And the best part was that it was an actual, real-life problem! Hence The Great Applied Problem was born. A couple of days later I presented this problem to my Precalculus students and asked them to solve it. It was a wonderful journey through mathematics involving these varied mathematical concepts: the Pythagorean theorem, area of a triangle, area of a sector of a circle, area of a segment of a circle, right triangle trigonometry, area of a circle, volume of a cylinder, volume of a non-standard prism, and several instances of units conversion. The solution of the problem also requires the student to organize his/her work well and to be able to logically develop a plan of problem solving.

THE TYPICAL AUDIENCE

The typical audience is a geometry class that has finished right triangle trig, area, and volume. I have been told by supervisors that this problem could be used for a final exam in a geometry class. I have used this problem in my precalculus class as a review problem.

MY GAME PLAN

I put the students into groups of 3, 4 or 5, whatever is convenient. I pass back the statement of the problem. I tell them to go ahead and solve the problem. I then walk around the room and listen, observe, answer clarifying questions, and give encouragement as possible. I don’t give hints yet. I really want the students to solve this problem without help from me.

After about 5 to 10 minutes, I do bring the class together to summarize what the groups have discovered and would like to share. One of the most common misconceptions is that students think that since the diameter of the tank is 14 feet, and the depth of the water is 4 feet, then the volume of the water is 4/14, or 2/7 of the total volume of the tank. This is a learning moment for many students. I do not tell the students that this is wrong. I prefer to have them explain it to each other and decide not only is it wrong, but WHY! This is a very powerful lesson.

Another very powerful learning moment is when students need to convert units – especially cubic feet into cubic inches. Many students want to multiply the number of cubic units by 12 to convert to cubic inches. Again I prefer that they decide among one another why this is terribly false.

From time to time, I do bring the class together and discuss what the students have discovered so far. By the end of the 45-minute period, most groups have a very good plan of how to solve the problem or are almost finished solving it.

After the group is finished solving the problem, the students can ask me if their answer is correct. They are absolutely thrilled when they are correct. If they are wrong, I try to point out where they may have made a mistake. The assignment for the first day is to finish the solution to this problem.

At the beginning of the second day, we discuss a plan of action. I put up a transparency of my solution (which appears later in this document). After this is discussed and alternative plans are heard, I give each student his/her own Great Applied Problem. A similar problem to the one we just did, but with different data. This individual problem is usually on the back side of the original problem in order to save paper. The directions are that students can work with whomever they wish. But ALL WORK MUST BE SHOWN neatly. See the NOTE on the bottom of the problem sheet. Communication, clarity, neatness, and organization will be used as assessment factors. The idea is that students can work together but no one can blatantly copy some one else’s work. This activity incorporates students working collaboratively and also working alone.

Since each of my 60+ students is getting a different set of data, does that mean that I have to make 60+ different answer keys? Absolutely. But don’t worry. I did it on a spreadsheet. I programmed the spreadsheet to compute every single intermediate calculation (in both feet and inches) so that I can properly assess their work and tell the student if a mistake was made, where that mistake was made. Otherwise it would be an overwhelming task for the student to find the mistake(s) because there are so many different computations. For that reason I allow the students to C&R – Correct and Return. I give them a second chance.

WHAT IS INCLUDED IN THIS PACKET?

1 –2 This two-page introduction
3 A page with the statement of the problem (same for all students)
4 A page with blanks to be filled in – individual problems
5 A page with the solution that I did by hand
6 – 8A 3-page solution of one individual problem turned in by one of my students
9 A page that explains the actual formulas given in the Excel spreadsheet
10 A page that gives a solution to this problem using Calculus

11–12A two-page document in Excel that has the solutions to 68 individual problems

including all intermediate results for each problem

Also included is a 15 page solution to this problem using Cabri geometry

software.

EXTENSIONS

I have presented this problem at many math conferences and institutes and have received much feedback. One extension that was told me was that the students realized that this tank was probably a gas storage tank for a gas station. So they made a pole that had markings on it. The markings would tell how many gallons of water (gas) was in the tank based on the depth of the water (gas). If you have any comments or ideas, please do not hesitate to contact me. Enjoy!

Tom Reardon (330) 757-4321

“GREAT APPLIED PROBLEM”

A cylindrical tank is lying horizontally on the ground.

Its diameter is 14 feet.

Its length is 20 feet.

The depth of the water in the tank is 4 feet.

One gallon is equivalent to 231 cubic inches.

A) How many gallons of water are in the tank?

B) How many more gallons of water will it take to fill the tank?

NOTE: In finding the solution to this problem, clearly communicate on paper how you are solving this problem so that anyone can follow your thought process. Neatly organize your work. Your grade will be based on organization, clarity, neatness, and communication as well as being correct.

C 2001, 1988 Reardon Gifts, Inc.

“GREAT APPLIED PROBLEM”

A cylindrical tank is lying horizontally on the ground.

Its diameter is 14 feet.

Its length is 20 feet.

The depth of the water in the tank is 4 feet.

One gallon is equivalent to 231 cubic inches.

A) How many gallons of water are in the tank?

B) How many more gallons of water will it take to fill the tank?

C 2001, 1988 Reardon Gifts, Inc.

“GREAT APPLIED PROBLEM”

Problem Number ______

A cylindrical tank is lying horizontally on the ground.

Its diameter is ______feet.

Its length is ______feet.

The depth of the water in the tank is ______feet.

One gallon is equivalent to 231 cubic inches.

A) How many gallons of water are in the tank?

B) How many more gallons of water will it take to fill the tank?

C 2001, 1988 Reardon Gifts, Inc.

“GREAT APPLIED PROBLEM”

Problem Number ______

A cylindrical tank is lying horizontally on the ground.

Its diameter is ______feet.

Its length is ______feet.

The depth of the water in the tank is ______feet.

One gallon is equivalent to 231 cubic inches.

A) How many gallons of water are in the tank?

B) How many more gallons of water will it take to fill the tank?

C 2001, 1988 Reardon Gifts, Inc.

GREAT APPLIED PROBLEM Reardon’s Solution by hand

C 2001, 1990 Reardon Fun Gifts, Inc.

Student Solution to The Great Applied Problem: Kim Kenzie Superb Effort

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C 2001 Reardon Fun Gifts, Inc.

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C 2001 Reardon Fun Gifts, Inc.

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C 2001 Reardon Fun Gifts, Inc.

THE GREAT APPLIED PROBLEM

Explanation of the formulas used in the Excel spreadsheet

ColumnExplanationFormula typed in row 8

ERadius in feet=B8 / 2

FHeight of triangle in feet=E8-D8

GBase of triangle in feet=2*SQRT(E8*E8-F8*F8)

HBase of triangle inches= G8*12

Area of triangle sq ft=.5*G8*F8

JArea of triangle sq in=8*144

KCentral Angle degrees=2*ACOS(F8/E8)*180/P()

LPart of circle=K8/360

MArea of circle sq ft=P()*E8*E8

NArea of circle sq in=M8*144

OArea of Sector sq ft=L8*M8

PArea of Sector sq in=O8*144

QArea of Segment sq ft=O8-8

RArea of Segment sq in=Q8*144

SVolume Water cu ft=Q8*C8

TVolume Water cu in=S8*(12^3)

UVolume Water Gallons A=T8/231

VVolume Tank cu ft=P()*(E8^2)*C8

WVolume Tank cu in=V8*(12^3)

XVolume Tank Gallons=W8/231

YGallons Needed B=X8-U8

C 2001, 1997 Reardon Fun Gifts Inc.

CALCULUS SOLUTION TO THE GREAT APPLIED PROBLEM

Place the circular side of the tank on

a coordinate system as shown to

the right:

C 2001, 1997 REARDON FUN GIFTS, INC.