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Name:___________________________________ Date:_________________________ Band:________

Calculus | Packer Collegiate Institute

Table of Contents:

1. Introduction to the Project; Where is the Calculus?; Your Grade (page 2)

2. Constraints on your Roller Coaster; Contest! (page 3)

3. Brainstorming Roller Coaster I (page 4)

4. Sketching Ideas for Roller Coaster I (pages 5 and 6)

5. Building a Geogebra Model of Roller Coaster I (page 6)

6. A Model of Roller Coaster I (page 7)

7. Data from Roller Coaster I (page 8)

8. Finding the Fun, the Cost, and the Bang For The Buck Factors from Roller Coaster I! (page 9)

9. Reflection on Roller Coaster I (page 10)

10. Brainstorming Roller Coaster II (page 11)

11. Sketching Ideas for Roller Coaster II (pages 12 and 13)

12. Building a Geogebra Model of Roller Coaster II (page 13)

13. A Model of Roller Coaster II (page 14)

14. Data from Roller Coaster II (page 15)

15. Finding the Fun, the Cost, and the Bang For The Buck Factors from Roller Coaster II! (page 16)

16. Reflection on Roller Coaster II (page 17)

2

You are someone who works for a roller coaster company (you and your partner need to pick the name of the

company):

You guys are going to design two strata roller coasters. There are only two of them in the world and you want

to bring more!

Your first coaster design is going to focus on being as fun as possible (given a few constraints)

Your second coaster design is going to try to be a bit more economical so you can sell it the plan to an

amusement park so they can get more “bang for the buck.”

But both are going to be strata roller coasters! WHOOSH!

You may be wondering how this relates to calculus. For now, we’d recommend suspending that question. We

promise there are connections, but you’ll see them as we go along. Instead, we suggest just having fun with it,

and trying to make the best roller coasters you can!

For those who need a little bit of foreshadowing, here’s the calculus…

There is a need to find the slopes of tangent lines to the curve in this project, but you also are going to be

analyzing functions (using geogebra). You’re going to study the shape of your roller coaster function. You’ll be

finding maximums and minimums, and when the slope is the most negative. At the end of the project, you are

going to relate these features of your roller coaster to the derivative function. (After the project, we’ll soon be

talking formally about how to relate the graph of a function to the graph of its derivative.) Additionally, we will

be hinting at some things we will be encountering in the second semester. Most notably how to find the

length of curves that are not straight lines (like, how do you find the length of track for your roller coaster),

and also something fancy called Riemann sums.

This project will comprise 20% of your grade (80% of your grade will be your skills). This packet will be graded

for correctness, neatness, and creativity. Take it seriously!

3

There are some constraints that are going to limit your roller coaster – to make it as realistic a roller coaster

project as possible.

Your roller coaster must start at ft0 x and end at 300 ft0 x .

When you board the roller coaster, the height of the roller coaster cannot be more than 30 ft . Imagine

riders having to climb more than 3 stories of steps just to get on the roller coaster. Anything more than

that is no good.

For similar reasons, when you exit the roller coaster at the end, the height of the roller coaster cannot be

more than 30 feet.

Your roller coaster cannot go underground. Although probably a cool thing, no amusement park owner

is going to be okay with the extra cost of building the tunnel.

Your roller coaster cannot go above 500 ft. It becomes a safety/cost issue if it does.

Your roller coaster track cannot end looking like:

Why? Because imagine if you rode that roller coaster and it

ended that way? CRASH! Either you’d be mangled… or if

somehow you got the cart to stop, you’d have a dickens of

a time leaving!

It has to look like something more gentle…

Thus, the end of your track can have a slope of -1/2, but nothing more negative!

There are two different roller coasters you are going to be designing. One that is meant to maximize fun and

one that is meant to maximize your bang for the buck. The contest is going to work as follows:

Prize for the first roller coaster design: If your group satisfies the constraints and gets the most fun

roller coaster, you will get an extra credit of 5% to your grade on this project. (This is among all groups

in all the calculus classes.)

Prize for the second roller coaster design: If your group satisfies the constraints and gets the most

“bang for the buck” roller coaster, you will get an extra credit of 5% to your grade on this project. (This

is among all groups in all the calculus classes.)

4

Below you and your partner are brainstorm ideas for things you want in your roller coaster that maximizes the

“fun factor” that we talked about in class.

max slope max slope peak 1 height peak 2 height length of track going down

total length of trackfrom peak 1 from peak10,000

2...

500 500Fun

Notes/Thoughts Below

5

Using your brainstorming, come up with rough sketches for what you want your first coaster to look like.

Recall you want this roller coaster to maximize the fun. Label things on the sketches that you think are

relevant (e.g. heights of things).

Sketch A

Sketch B

Sketch C

6

You are going to have to soon build a model of the most fun roller coaster. Looking at your three brainstorms,

which of the three sketches do you think will be the most fun (as measured by the “fun factor”)?

Sketch A Sketch B Sketch C

Why? What about that sketch, more than the others, makes it seem like it will maximize the fun factor?

Now you are going to use geogebra to build a model of your roller coaster that you thought would be the most

fun. This is going to be tricky – because not only do you have to satisfy the constraints, but you need it to look

as close to your idea as possible. If you can’t get it super perfect (if you had modeled too many humps), that’s

okay. Try to match the spirit/vision of your roller coaster as much as possible.

Take your time on this. This is the most time consuming and crucial thing you are going to be doing.

It also may be a bit frustrating. But keep at it. It will be worth it in the end.

Once you are done, save your roller coaster as: “Roller Coaster One, Shah, Lurain” (but using your own last

names, obvi).

On next page, attach a full-page printout of your geogebra model. Also, email your geogebra file to your

teacher, with subject line matching your file name.

7

Remove this page and attach a printout of your

geogebra model.

Things to look for:

o A clever title for your roller coaster! o The printout is a full page o The printout includes axes (with numbers labeled)

and gridlines o The points used to model the roller coaster are

hidden… the model is uncluttered. o The coaster starts at a height between 0 and 30 ft o The coaster ends at a height between 0 and 30 ft o The coaster is a strata roller coaster o The coaster does not exceed a height of 500 ft o The coaster does not go underground o The coaster does not end with a slope more

negative than -1/2

8

Remove this page and attach a page with the data on

your roller coaster neatly displayed in whatever way

makes the most sense for you.

Clarity in the presentation of your data is of the highest

importance.

Data to display:

o The height of the roller coaster at the peaks (and the x-value at the peaks)

o The height of the roller coaster at the valleys (and the x-value at the valleys)

o The maximum negative slope after each peak (and the x-value where the slope is most negative)

o The intervals (x-values) where the roller coaster is going up and the length of the track on each of those intervals

o The intervals (x-values) where the roller coaster is going down and the length of the track on each of those intervals

o The total amount of track o The total amount of track going downwards o The total amount of bracing needed to support the roller

coaster o The height of the roller coaster at the starting position o The height of the roller coaster at the ending position o The slope of the roller coaster at the ending position

9

Here you are going to calculate the fun factor, the cost factor, and the bang for the buck. Show your work.

FUN FACTOR:

COST FACTOR:

BANG FOR THE BUCK:

10

Remove this page and attach a page with your typed

answers to these three questions.

1. Is your roller coaster more fun than the example roller coaster we created in class? Why do you think it was (or wasn’t)?

2. Now that you’ve come up with this roller coaster, how could you tweak this design to come up with a similar roller coaster with a higher fun factor? Explain how your tweaks would increase the fun factor.

3. Look at your two unused brainstorming drawings. Now that you’ve actually modeled this one and gotten numbers, do you think one of the other two might be more fun? Why?

11

You want your second roller coaster to have more bang for the buck. If you have an infinite amount of money,

you can build pretty spectacular roller coasters. But at some point, cost becomes an issue. For this project,

let’s assume the cost of a roller coaster is how much metal you need to construct it (we’re assuming the

construction costs are negligible for this project).

Cost LengthOfTrack LengthOfBraces

If you’re pitching your roller coasters to amusement parks, to get them to build it, not only do you want to be

able to share with them how fun your roller coaster is, but also about how much it will cost to buy the

materials for it. Additionally, if you divide the fun factor by the cost, we get:

FunBangForTheBack

Cost

When we divide fun by cost, we get how much fun (bang) we get per unit cost (buck).

Question: If you were building a roller coaster without any consideration of how fun it is, but you want it to

still satisfy the constraints we talked about, what would the cheapest roller coaster look like?

Below, write down some thoughts about how you could create a fun roller coaster that gives you the most

“bang for the buck” (still satisfying the constraints we talked about).

12

Using your brainstorming, come up with rough sketches for what you want your second coaster to look like.

Recall you want this roller coaster to give you the highest bang for the buck. Label things on the sketches that

you think are relevant (e.g. heights of things).

Sketch A

Sketch B

Sketch C

13

You are going to have to soon build a model of the most “bang for the buck” roller coaster. Looking at your

three brainstorms, which of the three sketches do you think will be the best?

Sketch A Sketch B Sketch C

Why? What about that sketch, more than the others, makes it seem like it will maximize the bang for the

buck?

Like previously, you are going to use geogebra to build a model of your roller coaster that you thought would

yield the most bang for your buck.

Once you are done, save your roller coaster as: “Roller Coaster Two, Shah, Lurain” (but using your own last

names, obvi).

On next page, attach a full-page printout of your geogebra model. Also, email your geogebra file to your

teacher, with subject line matching your file name.

14

Remove this page and attach a printout of your

geogebra model.

Things to look for:

o A clever title for your roller coaster! o The printout is a full page o The printout includes axes (with numbers labeled)

and gridlines o The points used to model the roller coaster are

hidden… the model is uncluttered. o The coaster starts at a height between 0 and 30 ft o The coaster ends at a height between 0 and 30 ft o The coaster is a strata roller coaster o The coaster does not exceed a height of 500 ft o The coaster does not go underground o The coaster does not end with a slope more

negative than -1/2

15

Remove this page and attach a page with the data on

your roller coaster neatly displayed in whatever way

makes the most sense for you.

Clarity in the presentation of your data is of the highest

importance.

Data to display:

o The height of the roller coaster at the peaks (and the x-value at the peaks)

o The height of the roller coaster at the valleys (and the x-value at the valleys)

o The maximum negative slope after each peak (and the x-value where the slope is most negative)

o The intervals (x-values) where the roller coaster is going up and the length of the track on each of those intervals

o The intervals (x-values) where the roller coaster is going down and the length of the track on each of those intervals

o The total amount of track o The total amount of track going downwards o The total amount of bracing needed to support the roller

coaster o The height of the roller coaster at the starting position o The height of the roller coaster at the ending position o The slope of the roller coaster at the ending position

16

Here you are going to calculate the fun factor, the cost factor, and the bang for the buck. Show your work.

FUN FACTOR:

COST FACTOR:

BANG FOR THE BUCK:

17

Remove this page and attach a page with your typed

answers to these two questions.

1. Does your second roller coaster get more bang for the buck than your first roller coaster? Why do you think it did (or didn’t)?

2. Now that you’ve come up with this roller coaster, how could you tweak this design to come up with a similar roller coaster with even more bang for the buck? Explain how your tweaks would increase the bang for the buck.