| MAT 294 The Mathematics of Change I |
Fall 2005 Assignments |
| Context |
Comments |
|
| Engine Velocity |
To get s as a function of θ, you need to drop a perendicular from the top of the triangle (at the intersection of the crankshaft and flywheel) to the dotted line. This breaks the length s into two pieces, s1 from the center of the flywheel to the perpendicular and s2 from the perpendicular to the piston. Then get the vertical distance as 2.625sin(θ) and s1 as 2.625cos(θ). Then get s2 from the pythagorean theorem. This gets you to the first equation in the write-up. | pdf |
| Blood Vessel System |
Well-explained. | pdf |
| Flying Gravel |
The equations at the top of
page 2 are correct, but need to be explained. These are looking at only
the vertical component of motion. The first constant is found by
setting the initial vertical velocity equal to Vy. The second is
found be setting the initial height equal to 0 (the rock is essentially
starting at ground level). Then they solve for the time there y=0 which
is when the rock hits the ground again. At the bottom of page 2, this
time is substituted into the equation for x. This
equation for x as a function of α is what they want
to maximize! On page 3, the top graph is x vs. α. You can see where the
maximum is graphically. The bottom graph is the height vs. time so the
rock travels for just over 1.4 seconds. It is unclear what the table
and graph on the last page represent. |
pdf |
| Maximum Sustainable Harvest |
For the graph, it appears they have used r = 5 and P = 18. Note that the graph plotted is of f (S), not H(S) which is what is being maximized. By plotting H(S), you could determine the maximum harvest for a given r and P. | pdf |
| Bird Migration |
Well-explained. |
pdf |