The Three Stooges, Conic Sections, Trigonometry, and Implicit Differentiation Robert Davidson and Bob Gardner Department of Mathematics and Statistics East Tennessee State University April 6, 2012 (The Kickoff of Three Stooges Week) Online at: http://www.etsu.edu/math/gardner/stooges/Stooges-Trig-2012.ppt Motivation for the Project Stooges are a great attention device it keeps students interest. Trig is very applicable even in the setting of the Three Stooges. Watching the Stooges should relieve any
form of intellectual anxiety. Students learn by doing (measurement, unit conversion, scaling factor, trig). The Three Stooges Moe Curly Larry The Three Stooges 1. As a group, they were in show business for almost
50 years. 2. Made 190 shorts with Columbia Pictures. 3. Had their third short, Men in Black (1934), nominated for an Academy Award. 4. Had 4 different people in the role of the third Stooge. 5. Were the first to lampoon Adolph Hitler, in You Nazty Spy (1940). Are the Three Stooges Still Relevant?
Zogby International conducted a poll of 1,213 American adults by telephone in July 2006. One question asked for the names of the Three Stooges and another asked for the names of the three branches of government. Those able to name the Three Stooges: 73% Those able to name the three branches of government: 42% Source: http://www.zogby.com/Soundbites/ReadClips.dbm?ID=13498 20th Century Fox will release The Three Stooges movie on April 13,
2012. A Bird in the Head The 89th Columbia Pictures Three Stooges film; this was released in 1946. This short is classic Stooges and has the boys wall-papering a room. We watch from 2:35 to 4:28 (1:53 total). R. Gardner and R. Davidson, Mathematical Lens: The Three Stooges Meet the Conic Sections, Mathematics Teacher, February 2012, pp. 414-418. Stooges Among Us edited by Lon & Debra Davis BearManor Media (2008)
like Moe, Larry was only five-feetfour-inches tall. (page 10) Students Insert Known Data c a= 54 b Unit Conversion and Conversion Factor 1 foot = 12 inches, so the conversion factor is 12 inches/1 foot =
12/1 c a= = 64 54 So: 5+4 = 5x12/1+4 = 60+4 = 64 b
Students Make Measurements on Worksheet c= 10.9 cm a = 7.3 cm b = 8.1 cm Students Set Up a Scaling Factor 7.3 cm = 64.0 Scaling factor: 64.0 in / 7.3 cm = 8.77 in/cm Students must be warned to distinguish between a scaling factor (like on a map) and a conversion factor.
Students Scale Their Measurements c= 95.6 a = 64.0 b = 71.0Board length = 95.6 Now we use the scene where the board falls on Moes head to introduce angles and trig functions.
Nice! Suppose the board hits Moe on the head at a point 6 inches from the end of the board. A Schematic Diagram What is the angle, q,, the board makes when it hits Moe on the head? What are the six trig functions of q,? q,
Students Use the Pythagorean Theorem cm = 95.6 6 = 89.6 q, am = 64 bm = The 62.7 Pythagorean Theorem Students Find q,
cm = 89.6 q, bm = Trig Functions and Inverse 62.7 Trig Functions am = 64 q, = 45.6o Introduce Coordinate Axes
y From: http://www.collider.com/ dvd/reviews/article.asp/aid/5899/ tcid/3 x Questions About Coordinates What are the coordinates of the top end of the board when it is standing vertically? What are the coordinates of the top of Moes head? What are the coordinates of the end of the board when the board hits Moe on the head?
What is the equation of the circle traced out by the end of the board? Questions Involving Angles Convert q, to radians. Use the conversion factor p radians /180o. Through what angle did the board travel, from when it was vertical to when it hit Moe? Give your answer in radians. How far did the end of the board travel from when it was vertical to when it hit Moe? What area did the board sweep out from when it was vertical to when it hit Moe? More Online
Online, you can find a copy of this PowerPoint presentation, the PowerPoint presentation we use in class to introduce the worksheet, the worksheet itself, our solutions to the worksheet, and a summary of student impressions of the in-class experience described today: http://faculty.etsu.edu/gardnerr/stooges-trig/stooges-trig-2012.htm Now a more analytic approach R. Gardner and R. Davidson, Mathematical Lens: The Three Stooges Meet the Conic Sections, Mathematics Teacher, February 2012, pp. 414-418. Where does this ellipse
come from? Suppose the camera lens is located a horizontal distance d from Larry's feet and a vertical distance h. We will initially measure such distances in feet, but then convert them to units based on a coordinate system we introduce. h d We project the circle of radius r onto the plane through Larry's left foot and perpendicular to the floor. The radius from Larry's left foot to the point on the circle farthest from the camera projects onto a line segment of length y1. The radius from Larry's left foot to the point on the circle closest to the camera projects onto a line segment of length y2.
Plane of the Photograph h y1 r r y2 d-r d Now, we set up relationships between y1, y2, and other parameters of the
ellipse. First, set up a coordinate system in the plane of the photograph with the x axis horizontal, the y axis vertical, and the origin at Larry's foot. If Larry spins around on his left foot, the end of the board traces out a circle. In the plane of the photograph, this is an ellipse. y x The minor axis of the ellipse will then have length y1+ y2 and the center of the ellipse in the (x, y) coordinate system in (x, y) = (0, (y1- y2)/2). Plane of the Photograph
h y1 r r y2 d-r d Therefore the equation of the ellipse is + =1, y
y1 (y1- y2)/2 -y2 x or A+ B=1, where A = 1/a2 and B = 1/b2. y
y1 (y1- y2)/2 -y2 x We will measure distances in the studio (such as r, h, and d) in feet. We measure distances in the xy-plane in units such that the width of the photograph is 100 units. So we need a scaling factor s to convert r, h, and d into the units used in the xy-plane. s
Distances measured in feet. Distances measured in units. In the xy-plane, the point (rs,0) lies on the ellipse. We denote as (x0 , y0) the projection of the point at the end of the board when in its initial position as given in the photograph, onto the xy-plane. y (x0 , y0) y1 (y1- y2)/2
r -y2 rs x Since the point (rs,0) lies on the ellipse, then 1 + 2
2 2 ( 2 ) 2 =1.
(1) Since the point (x0 , y0) lies on the ellipse, then . (2) y (x0 , y0) y1 (y1- y2)/2
-y2 rs x Denote by m the slope of the x-axis in the xy-plane . Differentiating the equation of the ellipse with respect to x gives: + B = [1], 2A2B = 0. z x
y With x = x0 and y = y0, we have = m and so 2A2B (3) We now set up an equation for r using equation (1), (2), (3), and equations relating and to h and d. z x y
Equation (3) 2A2B (3) rearranges as: A==. Substituting this value of A into (1) 1 +
2 2 gives: 1 2 2 (
2 2 ) =1 1 (1) 2 2
2 2 0 + =1, 2 2 0 This equation can be solved for B and that value of B can be used to determine A: ( )
( ) = 4 0 2 2 2 (
1 2 0 2 ) + 0 (
1 2 2 ) and = 2 ( 1 2 2 0 )
2 2 2
( 1 2 2 0 )+ 0 ( 1 2 2 ) . Using these values of A and B in Equation (2) ( 2
0 ) 1 + 2 ( 0
2 2 ) =1 (2) ..... 2 2 2 2
2 2( 1 2 20)( 0 ) +0 ( 1 2 2 0 ) =2 ( 1 2 20 )+ 0 ( 1 2(4) ). gives Next, by similar triangles: h h 2 1 =
, = and + or h h = . and 2 =
+ 1 Therefore 2 h h 2 h 1 2=
= 2 . 2 + However, these relationships give the values of y1 and y2 in the same units as r, d, and h. We will adopt common units of feet and so need to convert them to the units used in the x y-plane. s Distances measured in feet.
Distances measured in units. 55.0 units 100 units In this plane, Larry measures as 55.0 units tall and we know that he is 5.33 feet tall. So we need to scale the values of y1 and y2 by an amount s = 55.0/5.33 units/foot = 10.32 units/foot. So in the units of the x y-plane, we have h h ,
= , 2= + 1 and 2 2 h
1 2= 2 . 2 Eliminating y1 and y2 in equation (4) gives 2 2 2 2
2 2 h 2h 2 2h 2h 2( ) 2 20 + 0 2 2 20 =2 2 2 20 2 2 ( ) ( ) ( )( ) 2
0 2 which, assuming , rearranges to the quadratic [ ][ 2 2 0 2 2 2 0 00 0 0 0 00 0 0
][ ] (h ) + (h )( ( ) )+ ( +h ) + ( ) =0 where we set From measurements in the photo, we have m = 0.22, x0 = -38.3 units, and y0 = 8.7 units. From above, s = 10.32 units/foot. The height of the camera (using the point at infinity) is roughly h = 4.5 ft. Height of Camera
The distance of the camera from Larry we estimate as d = 15 ft. With these parameters, the quadratic yields R = r2 = -41.43 ft2 and R = r2 = 35.06 ft2, and so . Finally, by the Pythagorean Theorem, the length of the board is l = ft = 7.97 ft = 95.60 in. l 5.33 ft r So howd we do? Presumably, we are dealing with a standard-length 8
foot = 96 inch board. If so, this means that our percentage error is: 0.4 96 100 %=0.42 % . Not bad for a couple of Knuckleheads! Any Questions? Thank You!
From: http://www.lunkhead.net/