Hot Air Balloons: Surfaces of Revolution

Pre-Calculus

This project is adapted from a project by Lynn Adsit and Kim Schjelderup Mercer Island Schools, Seattle WA

They adapted it from Robin Washam’s Workshop.

Hot Air Surfaces of Revolution A surface of revolution is a shape created by rotating a plane curve around its coordinate axis. For example, remember a standard cone is formed by rotating a line (𝑦 = 𝑥) around the x or y axis (we saw this briefly at the beginning of chapter 9). In our case, a balloon can be built by rotating the graph of a function about the x-axis.

It is possible to approximate the shape of a balloon by using functions that we have studied this year. For example, using the function 1

3 2

2

2

𝑦 = − (𝑥 − ) + 1 over the domain (0, 3) creates a region that is a rough estimate for a balloon. The shape can be modified by adding terms, multiplying x by a constant (horizontal stretches/compressions), multiplying the whole thing by a constant (vertical stretches/compressions), or even multiplying by another x (to make compound functions). An equation whose graph looks a bit like the outline of a balloon is: 𝑥

5

4

𝑥

𝑥

3

𝑥

2

𝑥

Use window:

Polynomial Functions: 𝑦 = 3 ( ) − 2 ( ) + ( ) − ( ) + 2 ( ) − 1 11

11

11

11

11

X: [0, 6] Y:[0, 3]

𝜋

Trigonometric Function: 𝑦 = 𝑥 sin ( 𝑥) where 𝑥 is in radians. 11

You will want to change that window to view your equations.

Both of those equations make a balloon shape, but they are a bit small for what we are doing.

Adding other functions, composites of functions, along with stretches/compressions, reflections, and translations can help produce a shape that looks even more like a hot air balloon. Creating a piece-wise function may also produce the shape that you are looking for. Conic sections can also be very helpful in making a balloon shape, especially when used in a piecewise function. In this project, you will choose a function whose graph resembles the cross section of a balloon. In order for the bottom to be open and to fit around the tube of the hot air source, the value of your function at 𝑥 = 0 must be at least 1 inch, so let’s say that at 𝑥 = 0, 𝑦 = 1.7". You will make the graph of the balloon cross section 12 inches wide and closed at the top, so that slightly beyond 𝑥 = 12 (the top of the balloon), 𝑦 = 0. Between those two points, 𝑦 > 0. In fact, to make the construction of the balloon easier, it would probably be a good idea to have a domain of 0 ≤ 𝑥 ≤ 12.5 and a range of 𝑦 ≥ 0. When you rotate the graph around the x axis, you will get the complete balloon shape. If you slice the balloon perpendicular to the xaxis, the cross section is a circle. The perimeter of the circle at any point is 2𝜋𝑦 (why?). To construct the balloon from tissue paper, you will divide the surface of the balloon into eight panels. At any point, each panel is approximately 1/8 th of the perimeter of the circle cross section at that point. To summarize, at each point 𝑥 along the graph of the function: The height of the curve is: The circumference of the circle cross section at that point is:

𝑦 = 𝑓(𝑥) 𝐶(𝑥) = 2𝜋𝑦

The width of the panel is:

𝑊(𝑥) =

𝐶(𝑥) 8

=

𝜋𝑦 4

The length of each panel is a bit more complicated than its width. Remember that you are covering the surface of the shape generated from a function that is not a straight line. If you go back to the original cross section, notice that the distance along the curve from any two points 𝑃(𝑥𝑃 , 𝑦𝑃 ) to 𝑄(𝑥𝑄 , 𝑦𝑄 ) is not necessarily the same as the difference in the 𝑥 values. If the two points are close enough together, a straight line is a reasonable approximation of the curve between the two points. You can calculate the length of the line segment from the two points 𝑃(𝑥𝑃 , 𝑦𝑃 ) to 𝑄(𝑥𝑄 , 𝑦𝑄 ) by using the distance formula: The distance from point P to point Q is:

2

2

𝑃𝑄 = √(𝑥𝑄 − 𝑥𝑃 ) + (𝑦𝑄 − 𝑦𝑃 )

Each Panel of the balloon can be approximated by a stack of trapezoids for intervals of 𝑥 from 0 to 12 in steps of about 1”. The result will be much like making a connect-the-dot drawing of the balloon panel. Each trapezoid has a bottom base with width 𝑊(𝑥𝑃 ), the top base (parallel to the bottom) width 𝑊(𝑥𝑄 ), and the length is the distance 𝑃𝑄. You want to make a tissue paper balloon large enough to fly with the hot air generated by a portable charcoal burner, so you will have to scale it up to make it 3 times the size of your graph model.

Requirements of Project 1.

Attendance = 10 points (Homework) – 2 points per day; make-ups for excused absences after school

2.

Mathematical Work = 40 points (Homework) Includes a. Including this packet completed thoroughly; b. Physical elements: cross section, panel, balloon; c. Meeting each mathematical requirement of the cross-section and pattern creating d. A 5 point penalty may be assessed for each teacher assistance with the mathematical process, so think through and discuss with your partner before you ask for assistance.

3.

Balloon Construction Process = 30 points (Quiz) Includes a. Following directions/meeting requirements b. Working collaboratively c. Making deadlines for the process/having the materials (tissue paper – 2-3 packages per group) on time

4.

Evaluation (test on process and key concepts) = 100 points (Test) You get to use: a. Your partner b. This completed packet

5.

Balloon Flies (gets above the ground for at least 5 seconds) = 10 points (Quiz)

Timeline for Project Date

Balloon Project

5/16

Introduce the project. Pick partner. Begin finding an equation.

5/17

Determine equation. Draw cross section. Calculate panel widths and lengths.

5/18

Create large panel pattern. Trace/cut panels.

5/19

Glue panels. Glue wire.

5/20

Glue panels. Glue wire.

Weekend 5/23 Testing Schedule

Periods 1-3 only. Work on balloons

5/24 Testing Schedule

Periods 4-6 only. Work on balloons

5/25

Finish math work in packet. Finalize balloons.

5/26

Fly Day! Fly the balloons (if weather permits)

5/27

Balloon Test

Steps for Designing a Hot Air Balloon Step 1: Equation Explore various types of equations of the functions (piece-wise, polynomial, sinusoidal, conic section, etc.) that could represent the outline of the balloon. - Starting equation: _________________________________________________________________________________ - Record the progressive alterations you made to your starting equation and why.

- Describe your method for choosing the “best” function. What were your criteria/thoughts?

Equation: ______________________________________________________________ Mr. Swanson’s Okay: ____________

Step 2: Scale Drawing of Cross Section Fill in the table with the values of your function at each 1 inch interval; i.e. find the y-value associated with the corresponding x-inch value for your function. An easy way to do this is to use a table or list on your calculator. In a list, enter the numbers 0 through 12 in the first list. In the top (equation line) of the second list, enter your equation. You will use these data to create your scale drawing. 𝐿1 𝐿2

Inch mark 𝑓(𝑥)=height of curve

0 1.7”

1

2

3

4

5

6

7

8

9

10

11

12 0”

Draw a scale drawing of the outline of the cross section. Remember that the mouth of the full sized balloon should have a diameter of approximately 3” so that it will fit over the hot air source. It will look something like this:

On a piece of graph paper, draw your balloon outline using the cross section equation values from the table. Glue this drawing to a file folder using the x-axis of the drawing as the fold line of the file folder. If you cut along the function line and then open the folder with a 45° angle between the sides, you will see 1/8th of your balloon. Hopefully this will help you to envision the shape of your balloon’s eight panels.

Step 3: Panel Widths Your balloon is a three-dimensional object yet you will be making it using two-dimensional materials: tissue paper. In order to do this, you will create 8 congruent panels which when glued together will form the three-dimensional balloon. A method of approximation that is useful is to use various circles stacked along the x-axis of the balloon and create the shape of the balloon. The circumference of each circle gives you the girth of the balloon at various points along the axis. You will need to figure out the circumference of the cross sectional circles. Calculate the circumference of each circle (to the nearest tenth), then calculate 1/8 th of the circumference.

𝐿1 𝐿2

Inch mark

𝐿3

Circumference 𝐶(𝑦) = 2𝜋𝑦

𝐿4

0

𝑦 = 𝑓(𝑥)= radius Of the circle

1

2

3

4

5

6

7

8

9

10

11

12

1.7”

≈ 10.7

Panel Width 𝐶(𝑦) 𝑊(𝑦) = 8

≈ 1.3" ►DO THIS NOW: Transfer the Panel Width values to the table on the next page.◄

Step 4: Length of Balloon’s Outer Edge = Panel Length The panel must fit over the edge of the cross sectional piece, not along the main axis of the balloon; therefore, it is necessary to find an approximation of the arc length between each inch mark on the cross section. Using these arc lengths as tic marks on a new x-axis, you will construct a panel making sure that the 1/8th circumference is measured at each corresponding tic mark.

In order to approximate the arc length, you will make trapezoids at each pair of tic marks. Using the ordered pairs, (inch mark, radius of circle), you will use the distance formula (see the first page) to calculate an approximate length of each arc.

Inch Mark 𝑦= 𝑓(𝑥) =radius Of the circle

0

1

2

3

4

5

6

7

8

9

1.7”

Arc Lengths using Distance Formula, 𝐴𝑛

Finally, add the arc lengths from the above table to determine the panel length: _____________________ ►DO THIS NOW: Transfer the Arc Length values to the table on the next page.◄

10

11

12

Step 5: Constructing the Panel Pattern Remember, you want to make a tissue paper balloon large enough to fly with the hot air generated by a portable charcoal burner, so you have to scale it up to make it three times the size of your graph model (besides, who wants a smaller balloon anyways?). Then take ½ so you can measure from the center axis along each tic mark 𝐴𝑛 . Do this now in the table below. Inch marks Arc Lengths, 𝐴𝑛 , using Distance Formula

0

1

2

3

4

5

6

7

8

9

10

11

12

Scaled Arc Lengths 3 ∙ 𝐴𝑛 Panel Width 𝑊(𝑦)

Scaled Panel Width 1 ∙ 3 ∙ 𝑊(𝑦) 2

≈ 1.3"

≈ 2.0"

Panel Length: _____________________

Full size panel = 3*panel length: _______________________

Step 6: Creating the Panel Pattern Take a piece of butcher paper that is a little longer than the full size panel and fold it in half. Make sure the folded width of the paper is wider than the largest width you found in the table above. Use a strip of graph paper along the fold of the butcher paper to measure each arc length. Mark off the arc lengths along the folded edge. Be careful!! You will count off each new arc length from the previous tic mark.

Next, at each tic mark, measure perpendicular to the folded edge the calculated scaled panel widths. Connect the endpoints of these lengths with a smooth curve and cut out the pattern for the panel (this is called a gore).

Step 7: Constructing the Balloon Cut out the eight panel pieces at the same time… these must be exactly the same size and shape. Use the paperclip method shown in class. The seam allowance should be about 1/2” beyond your pattern. You will then glue the panels together in the manner we talked about in class. It is critical that you use a thin bead of glue smoothed out with your finger; too much glue will cause the panels to stick (and dry) together. Also, use wire at the mouth of the balloon. Roll it up in the tissue paper and then glue. This will help keep the mouth open while getting the hot air. Put your names, and the school name (SHS) and the school phone number (360-629-1300) on the balloon so if it flies away, it might be returned!

Day of Balloon Flight Directions Final Touches on Balloon Please return to: Stanwood High School 360-629-1300 Optional: include your names

Organize Balloon Packet 1. Names on packet 2. All sections completed a. calculations neat and complete b. explanations written and complete 3. Cross section a. both names on it b. equation on it 4. Panel pattern a. both names on it b. folded neatly to the size of 8.5” x 11” (a normal sheet of paper)

Hot Air Balloon Flight 1. Get balloon and walk quietly to the designated flying area. 2. Work with your partner only. 3. The person holding the balloon over the chimney MUST wear the gloves. The other person ‘fluffs up’ the balloon as it fills with hot air. 4. When ready, hold your cross section up next to your balloon to determine how well your constructed balloon matches your plan. 5. Let the balloon go. 6. You and your partner are responsible for collecting your balloon when its flight is done.

Hot Air Balloon Evaluation The next class period you will take your evaluation on this project. 1. Work with your partner only. 2. Use your packet as needed. 3. Answer questions completely but concisely. When you are done with the evaluation you will staple it to the front of your packet (with the cross section and panel pattern attached – see above) and turn it in.

Hot Air Balloon Project Scoring Names: _____________________________________________________ Period: _________ Partner’s name: _____________________________________ Attendance: 10 points (2 points each, up to 10 points) Homework Tues 5/17

Wed 5/18

Thurs 5/19

Fri 5/20

M/T 5/23-24

Wed 5/25

Fly Day

Yes/no Make-up Day

Total: _________

Mathematical Work: 40 points Homework Equation accepted Scale drawing of cross section Panel widths Length of balloon’s other edge = panel length Scaled lengths

5 pts - ________ 5 pts - ________ 10 pts - _______ 10 pts - _______ 10 pts - _______ Total: _________ Total Homework score: _______/50

Balloon Construction Process: 30 points Quiz Follows directions Works well with partner (sharing the load) Meeting the process deadlines/having materials on time Balloon Flies: 10 points Quiz At least 5 second in the air

10 pts - ________ 10 pts - ________ 10 pts - ________ Total: _________ 10 pts - ________ Total Quiz Score: ___________/40

Evaluation: 100 points Test

Evaluation Score: ___________/100

Hot Air Balloon Project Scoring Names: _____________________________________________________ Period: _________ Partner’s name: _____________________________________ Attendance: 10 points (2 points each, up to 10 points) Homework Tues 5/17

Wed 5/18

Thurs 5/19

Fri 5/20

M/T 5/23-24

Wed 5/25

Fly Day

Yes/no Make-up Day

Total: _________

Mathematical Work: 40 points Homework Equation accepted Scale drawing of cross section Panel widths Length of balloon’s other edge = panel length Scaled lengths

5 pts - ________ 5 pts - ________ 10 pts - _______ 10 pts - _______ 10 pts - _______ Total: _________ Total Homework score: _______/50

Balloon Construction Process: 30 points Quiz Follows directions Works well with partner (sharing the load) Meeting the process deadlines/having materials on time Balloon Flies: 10 points Quiz At least 5 second in the air

10 pts - ________ 10 pts - ________ 10 pts - ________ Total: _________ 10 pts - ________ Total Quiz Score: ___________/40

Evaluation: 100 points Test

Evaluation Score: ___________/100