April 18, 2007
*Ball floating in mercury problem.
*D is the area of the “slant triangle.”
This gives an analog of the Pythagorean Relationships.
Note the extensions via rays in the following part of the same figure:
Show that D and C are equidistant from A.
*Note: Stewart’s Theorem gives the above relationship regardless of whether or not the “t” segment is an angle bisector. (The proof requires using triangle areas).
A man walked for five hours, first along a level road, then up a hill, then he turned around and walked back to the starting point along the same route. He walks 4 mph on the level ground, 3 mph up the hill, and 6 mph downhill. Find the distance walked.
Claim: When you get a solution for this, if you change the numbers above, you’ll not get the solution.
Not hard to show that this must be a right triangle via algebraic manipulation, but how could you show it geometrically?
S1 = 1 + 2 + 3 + … + n
S2 = 12 + 22 + 32 + 42 + … + n2
S3 = 13 + 23 + 33 + … + n3.
Show that S3 = S12.
Find the length of the curve that passes through each of the three points:
If 6 acres of grass, together with what grows on the 6 acres during the time of grazing, keep 16 oxen for 12 weeks, and 9 acres keep 26 oxen 9 weeks, how many oxen will 15 acres keep 10 weeks, the grass growing uniformly all the time?
Grass Consumption by Oxen Problem
a = amount of grass per acre at start
g = amount of grass per acre that grows per week.
x = amount of grass eaten by one ox per week
amount eaten plus amount that grows back = amount of grass 16 oxen eat in 12 weeks
6a + 12*6g = 16*12x
9a + 9*9g = 26*9x
15a + 10*15g = y*10*x
Divide through by x (because these are rates):
6k + 12*6m = 16*12
9k + 9*9m = 26*9
15k + 10*15m = 10y where k = a/x and m = g/x