Now that the lizards (Frank and Louie) have stolen the spotlight, Bud, Weiss and Irv are ready to move out of the old Pad. Unfortunately, the Pad is designed to keep all of its inhabitants in, and visitors out. Nonetheless, on a late night hop through the bog, Irv discovered a secret passageway, believed to lead to freedom! The escape route is a tunnel running underneath the swamp, with a triangular passageway, or door, midway through the tunnel. See the image on the left below.
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. Your assignment is to derive a formula for the curve of doom, say d(x) (using calculus, derivatives and properties of maximums). This equation is needed to help Bud, Weiss and Irv escape (they will attempt to jump over the curve of doom). Quite obviously, the curve's height 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 xb = h, xw = h + 5 and xi = 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. For extra credit (up to 15 points) determine a value of h (in inches) so that all 3 clear the curve of doom if Bud jumps from xb = h inches, Weiss from xw = h + 5 inches and Irv from xi = h + 10 inches (these locations are the centers of their bodies). Frank and Louie (the lizards who took over Bud, Weiss and Irv's acting careers, and should not be blindly trusted) suggest that you derive your formula for d(x) in the following manner. Let bt(x) denote the equation of the line of the laser beam when the receptor has an x-coordinate equal to t. Then (x, y) is on the curve of doom if and only if y is the maximum (over all values of t from 0 to 1) of the heights bt(x). Thinking of x as fixed, d(x) is then a maximal value (with respect to t), namely the largest y value above x, on any of the laser beams (notice how the red curve above sits just atop all of the laser beams). You might also get some graph paper and draw in several laser beams, and estimate various values on the curve of doom, to get a feel for what is going on. 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 derived 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 (or any other simple curve), as nothing in this handout says that it is a parabola! Plain and simple, points on the curve of doom come from maximizing y coordinates on the laser beams.