The door has a
height (measured vertically from the bottom vertex of the triangle, straight
up to the roof of the triangular doorway) of 4 feet. The top of the doorway
is 8 feet wide, and the bottom vertex is exactly in the middle (so it's
4 feet horizontally from the bottom vertex, to a point, directly below
the upper right, or upper left corners of the doorway. To prevent persons
(as well as frogs, lizards, varmints, etc.) from navigating through this
tunnel, the owners of the Pad have installed an extremely sensitive and
lightning fast laser mechanism in the left and right diagonal edges of
this doorway. In the left diagonal edge of the triangle, there is a moving
laser beam generating device. It slides (several times per second) smoothly
from the bottom vertex of the triangle to the top left vertex and back
again, repeatedly. Simultaneously, a laser beam receptor is embedded in
the right diagonal edge of the triangular doorway. The receptor slides
(at the same lightning fast speed as the emitter) from the bottom vertex
to the upper right vertex, back to the bottom vertex, and so on.
Imagine placing a set of axes over the picture of the doorway above,
so that the origin is at the bottom vertex of the triangle, and the left
and right diagonal edges of the door lie on the graphs of y = -x and y
= x respectively. Furthermore, scale the x and y axes so that one unit
(in both the x and y directions) corresponds to 4 feet. The triangle shown
above then has vertices at (-1,1) (top left), (0,0) (the bottom) and (1,1)
(top right). 
The
laser beam receptor now runs along the graph of y = x, from x = 0 to x
= 1, while the emitter runs along the graph of y = -x, from x = 0, to x
= -1. These two devices move in such a way that when the x-coordinate of
the receptor is equal to t, the x-coordinate of the emitter is t - 1 (for
t between 0 and 1). See the image to the left, where a variety of laser
beams are shown. Below, you can see the motion of the laser beams (slowed
down by a nearly infinite factor) as the emitter and receiver (not shown
in the image, and embedded inside the blue wall of the doorway) travel
along the diagonal edges of the doorway.
The motion of the laser
beam emitter and receptor determine a Curve of Doom (the
red curve in the image to the left), below which, no mentally stable frog
would choose to pass. Quite obviously, the curve is minimal above x = 0,
but Bud, Weiss and Irv need to jump through this terrible triangular doorway
at exactly the same time (what happens if they don't is not fit for print).
Bud, Weiss and Irv are all exactly 5 inches wide. Furthermore, all three
have an uncanny ability to jump in such a way that only the exact middle
of their bodies is in danger of falling below the curve of doom, but they
must take off with their midsections aligned side-by-side, meaning at a
triple of locations x = h, h + 5 and h + 10 inches, with h between -45.5
inches (or roughly -0.947917 units) and 35.5 inches.
The swank style of living in the bog has hindered the frogs jumping abilities. Bud can clear 24.1 inches (roughly 0.50208 units) with a vertical leap, Weiss can handle a jump of 24.05 inches and Irv can clear 24.6 inches. The three frogs are quite stubborn when it comes to order, they always proceed with Bud on the left, Weiss in the middle and Irv on the right. Your task is to determine a formula for the curve of doom, and for extra credit (up to 15 points) determine a value of h (in inches) so that if Bud jumps from x = h inches, Weiss from x = h + 5 inches and Irv from x = h + 10 inches (these locations are the centers of their bodies), all 3 clear the curve of doom. Frank and Louie (the lizards who took over Bud, Weiss and Irv's acting careers, and should not be blindly trusted) suggest that you consider the horizontal distance t, of the laser receptor from x = 0, as a parameter, and for each fixed value of x (from negative one to one), try to maximize (with respect to t) the y coordinate above x, that lies on the laser beam which is received by the receptor at x = t. If you connect these maximal y-coordinates with a curve, you'll get the curve of doom (notice how the red curve above sits just atop all of the laser beams). You could also get some graph paper and draw in several laser beams, and estimate various values on the curve of doom. Your formula for the curve of doom should give values close to these estimates. Of course, the majority of your grade will be based on your ability to clearly explain how you found the formula for the curve of doom, to a person who has never read this document. The reader should be able to reproduce your steps and calculations, completely from your write-up. You should not assume that the curve of doom is a parabola, as nothing in this handout says that it is a parabola!
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