Naturally, that caught my attention - a geometry PoW! How convenient. So let's figure this out. Is 31 degrees steeper than the roof on my house? First problem is that I don't know how steep my roof is. However, I do know how steep the roof is on the house I helped build when I was in college, so let's use that.

The main house roof had a 10-inch pitch. This means that the roof goes up 10 inches for every 12 inches of horizontal measurement. (Remember slope from algebra?) Is the Daytona speedway track steeper than this roof?

How about the garage roof on the house I built? That roof had an 8-inch pitch.

Make sure you draw a good picture. You might have to look up how to figure out the angle of each roof, but I think you can do it!

This week's problem proved to be a review for some folks, and a look ahead for others. I didn't think that everyone had done trigonometric ratios yet, but it doesn't hurt to have to look them up, either - in fact a number of you said that you did just that.

82 folks got it right this week, and 72 got it wrong. Many of those who got it wrong did so because they drew an "accurate" picture and measured it with a protractor. Now, almost all of these folks got the "answer" right, but I got picky. Measuring a drawing will rarely give you an accurate answer. You got lucky because both roofs were steep enough that even a drawing got you the right answer in most cases. I actually used the real numbers for those roofs. I could have made one of them a 7-inch pitch, and that would have been quite a bit closer to the speedway. Then those pictures wouldn't have worked. When there are tools available that will give you an exact answer, you should use them, and you will rarely, if ever, get credit for measuring a drawing. You should always figure that there's a really good way to figure out the answer, and try to find that. Yes, even if it means searching through your textbook or asking for a hint from someone who might know (parents and teachers come to mind).

Then again, maybe I should have just said, "no protractors allowed!" That would work. How would you find help on something like this in your book? Flipping through the pages would work. You might try looking in the table of contents for things that have to do with right triangles, too.

For some good examples, I've included three solutions. Brian Anderson and John Camp of Loveland High School drew a nice text picture and gave a good explanation. Aya Kohinaka of the American Embassy School in New Delhi drew a very nice picture depicting what "pitch" really means. Her classmate Matt Smawfield drew a super picture of each roof, as well as an illustration of the tangent ratio. Both of them also included good explanations of tangent and of how they tackled the problem.

When you are explaining how you did a problem like this, think about saying something beyond, "hit the tangent^-1 button on your calculator." Don't say what to do; explain what needs to be done - we need to find the inverse tangent. That'll show that you really understand what's going on.

Only one person was lazy enough to find the slope of the track (one operation) instead of the degree measures of both roofs (two operations). Maybe that is because we tend to be more comfortable comparing degrees than comparing slopes? I dunno.

Three people sort of gave the pitch of the track, which was neat. I say sort of because one person gave it exactly, but didn't say how he did it, and two other people guessed at numbers until they got something that came close. To do something like that, think about how you found the degrees of the roof given the pitch - divide by 12, and find the inverse tangent. So if you want to go from degrees to pitch, go backward - find the tangent, then multiply by 12.

I always appreciate comments that show that you're paying attention. More than a few people mentioned the fact that the announcer was wrong for my house, but that houses aren't all the same. Aaron Mertz and Alison Miller both found the pitch of their roof and compared it to the race track. Alison's house has a steeper pitch, and Aaron's house has a less steep pitch. Sara Clagg asked if roof pitch might be a function of climate. Steve Martin commented that most roofs in Florida (which is where Daytona is) have shallow roofs, probably because they have lots of hurricanes. I wrote to Sara that the house I built was in Vermont, and roofs there are typically steep so that the snow falls off them in the winter and doesn't collapse the houses.

A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.

From:   Brian Anderson and John Camp
        
Grade:  10
School: Loveland High School, Loveland, Ohio

     The turns on the track are not at as steep of an angle
as either roof. Here's why:

please, try to imagine that this is a slice of a roof.

                 |
             '   |10
        .        |
     ------------
    /\    12
    ||
  angle x

We need to determine whether angle x is greater or
less than 31 degrees.
In this slice of our roof. The base is equal to 12. 
Since the pitch of the first roof is 10 that means
that for this slice the other leg is equal 
to 10.  Because the tangent of
angle x is equal to 10/12 (opposite over adjacent),
we find which angle has a tangent of 10/12.
This angle turns out to be 39.805571092265.
Therefore roof 1 is steeper than the track.

Using the same method we can determin that the angle of 
roof 2 is equal to 33.69006752798, also greater than
the angle of the track.

From:   Aya Koshinaka
        
Grade:  9
School: American Embassy School, New Delhi, India

Subject: Feb 17-20

Dear Annie,
	Here is my solution.  Yes, the house and garage's roofs are steeper
than the car race's bank because I figured out the two roofs angles.  To
get the angle of roofs, I used tangent.  Tangent is the ratio of the length
of the opposite leg to the length of the adjacent leg of a right triangle.
So, for house's roof, I did 10/12, and for garage, I did 8/12.  Then I used
my calculator to find the angles and it said that house's roof is 39.8
degrees and 33.69 degrees for the garage roof.  The car race's bank is 31
degrees.  So, I can say that the two roofs are steeper.



	I am Aya Koshinaka.  I am in 9th grade in American Embassy School
in New Delhi, India.  I am doing this because my teacher said us to do this
for credit and I have already sent 3 or 4 of my solusions before.  I hope
my solusion is right.
	*E-mail address of American Embassy School is
			

From:   Matt Smawfield
        
Grade:  9
School: American Embassy School, New Delhi, India

Subject: Roof Angles

NAME: Matt Smawfield
E-MAIL:  (note: giasdl01 - that is L zero one)
SCHOOL: American Embassy School, New Delhi, India
GRADE: 9
The Corner Banking of the Daytona Speedway
February 16-20, 1998



To solve this problem, we have to make use of trigonometric ratio. A
trigonometric ratio, is the ratio of the lengths of two sides of a right
triangle. The particular ratio that we are going to use, is the tangent
ratio. The tangent ratio, is the ratio of the length of the opposite leg to
the length of the adjacent leg of a right triangle. 
               Length of opposite leg
TAN X= ------------------------------ 
               Length of Adjacent leg
With this formula, we can find the measure of the angle. 
If the opposite leg was 6, and the adjacent leg was 4, then we could say
that 6/4=TAN X,  TAN X=1.5. By pressing the TAN -1 button on the
calculator, we get a number around 56.3. Therefore, the measure of angle
X=56.3º
House: In this problem, we are given the inch pitches of two roofs. In the
first roof, there is a 10in pitch. What that means is that for every 12
inches of horizontal measurement, the roof goes up 10 inches. This then
gives us a right triangle, two legs, and an angle to solve for. 10 is the
opposite leg, and 12 is the adjacent leg. Therefore, to find X, you would: 
10/12=TAN X 
0.83=TAN X
X= TAN-1  0.83,
X=39.69º
Therefore the roof of the house is steeper than the slope of the racetrack.

Garage: The garage is a similar problem, and is solved the same way. The
problem is identical, but the difference is that there is an 8in pitch
rather than a 10in pitch. Therefore, the only thing that changes, is the
length of the opposite leg. Again we use the tangent to solve for the angle X:
8/12=TAN X
0.66=TAN X
X=TAN-1  0.66
X=33.42º

Therefore the garage roof is steeper than the slope of the racetrack.

Answers:
Race Track = 31º
House Roof = 40º - no the racetrack curves are not steeper than the roof.
Garage Roof = 33º - no the racetrack curves are not steeper than the roof.

The following students submitted correct solutions this week: