Some Math Questions?

Question

1. In Taxi cab geometry how is distance measured?
2. solve for pi in pi*d
11. Compute the circumference of the circle x2+y2=9 giving the exact value and the value to three significant digits
12. What is the value of pi you used to give the approximate value
14. find the circumference of the taxi circle (x2+y2=9)
15. Find the value of pi in taxi cab geometry
16. Will the value of pi always be the same value you computed in #15? Dont guess use at least one equation with a different radius and another equation with a different center and show work in your journal!

another question
10. In Words you could say from #2 that pi is the ratio of the _____ to the _____ in a circle

Answer 1

1. RPM (n) of the wheel is measured; Wheel dia (d) is known and time (t) in minutes for which the taxi has run is also measured.
Distance covered = pi*d*n*t

2. doesn't make any sense

10. circumference, diameter

11. radius = 3; circumference = 2*pi*radius = 2*3.14*3 = 18.8

12. 3.14

14. same as 11

Answer 2

First, here is my understanding of the taxicab metric: the distance D((x1,y1),(x2,y2)) from (x1,y1) to (x2.y2) is |x1 - x2| + |y1 - y2|. Geometrically, this is the sum of the lengths of the segments parallel to the x-axis and parallel to the y-axis which connect (x1,y1) to (x2,y2).

If that is not your definition of the t.c metric, ignore the rest of this.

1. This has been answered above.

Before proceeding, let us consider a fundamental question: What is the equation of the unit circle relative to this t.c. metric? The equation of the unit circle (circle with center at the origin, radius 1) in the Euclidean metric is the set of all points (x,y) such that distance from (0,0) to (x,y) = 1. So, if (x1,y1) = (0,0), the points on the unit circle in t.c.-land is the set of (x,y) satisfying |x| + |y| = 1. There are four possibilities: the points on the lines x + y = 1, x + y = 1, -x + y = 1, and -x - y = 1, all with the obvious restrictions that -1 < x < 1 and -1 < y > 1. Now we see that the unit "circle" is the square with vertices (1,0), (0,1), (-1,0), (0,-1).

2. The area of our unit circle is 1. The diameter (the largest "chord" which can be drawn) is 2. The circumference of our circle is 8 (the distance from (1,0) to (0,1) is 2, etc.), so if circumference is pi*d, we have 8 = pi*2, so pi = 4. But on the other hand, if area is pi*r^2, with area = 2^2 = 4 and r = d/2 = 1, we still get then pi = 4.

If this is not enough, come back at us.