Original Math Notes
All Seasons
Season 5
Episode 507: Magic Show
Blood spatter
Scene 6:
SHOT -- Charlie and Larry studying the blood under Aquarius.

                    CHARLIE
          The blood could be a prop, part of
          the act.  Spray some blood on the
          walls, vanish, and voila, got a
          pretty good trick.

                    DON
          And pretty good publicity.  So you're
          thinking it's a hoax?

                    CHARLIE
          Easy enough to figure out.
            (off their looks)
          We can use an application of blood
          spatter trigonometry to study the
          correlation of this blood to Talma's
          probable position within the box.

Blood Spatter Trigonometry »

A moving droplet of blood will eventually hit a surface. At the initial instance of contact, the spherical droplet will be tangent to the intercepting surface. The type of spatter the blood droplet leaves on the surface strongly depends on its direction and speed (its vector). In the diagram, the instance of contact is shown for the red spherical droplet. A Dandelin cone based on the vector approximates the elliptical spatter pattern the blood will leave behind. Note that the initial tangent point is one of the foci of the underlying ellipse. The speed and viscosity will further deform the elliptical pattern.
Scene 11:

                    CHARLIE (cont'd)
          We were able to analyze the pattern of
          the blood droplets, along with the
          angle and velocity of impact to
          determine, with high probability that
          Talma suffered a real injury.  It could
          have been accidental, or it could have
          been the result of a struggle...
            (takes out Penn's card)
          I'd like to consult a couple of experts
          who might have an idea which it was.

Linear Transformation with Given Eigenvectors »

Two eigenvectors and their corresponding eigenvalues (taken from the length of the vectors) can determine the deformation of a circle into an ellipse. The image shows how a linear transformation, uniquely determined by these eigenvector/eigenvalue pairs, transforms points on the unit circle.
Scene 25:
          
Charlie punches up the glass cylinder on the monitor.

                    LARRY
          Witnesses from the performance report
          that it was full.

Charlie presses a key, the cylinder fills.

                    CHARLIE
            (quick calcs in his head)
          Okay, we're looking at approximately
          50 cubic feet of water...

                    AMITA
          400 Gallons.

                    CHARLIE
          Water weight 3,300 Pounds.

                    AMITA
          A little over one and a half tons.

Volume of Cylinders »

This Demonstration shows the volume of a cylinder.
Scene 38:
Charlie's laying on his stomach where Aquarius would have
sat, digging around in the guts of the machinery.

                    CHARLIE
          I count three input pipes.

Larry brings up 'Figure 2 - Water System.'

                    LARRY
          Only three?  Reconstruction of the
          apparatus indicated at least four..

                    AMITA
          Six if they were playing it safe.

                    CHARLIE
          To fill the cylinder with three pipes
          in five seconds...

The Hydraulic Press »

A hydraulic press is a machine made of two pistons with different cross-sectional areas (A and a), which is used to lift large weights or compress material. The ratio of the distances moved and the ratio of the forces are both equal to the ratio of the areas of the pistons.
 
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