Show that every map on RP² can be colored by six colors, and that there is a map on RP² that requires six colors.

A company makes two kinds of animal food, A and B, which contain two food supplements. It takes 2 pounds of the first supplement and one pound of the second to make a dozen cans of food A, and 4 poundsof the first supplement and 5 pounds of the second to make a dozen cans of food B. On a certain day 80 pounds of the first supplement and 70 pounds of the second are available.   Use the geometrical approach tofind the number of cans of each food should be made to maximize profit, if the profit on a dozen cans of food A is $3 and the profit on a dozen cans of food B is $10. Show every necessary steps.

Give a example of a set of 4 points in r3 that are in general positions

You may use a map to answer this question. B = { x: x is a state that borders Illinois} Write B using the roster method

For each of the following pairs of points, find the length of AB. a. A(0,5), B(0,4) b. A(0,15), B(36,0) c. A( 19, 14), B( 24, 15)

Derive the linear interpolant through the two data points (1.0, 2.0) and (1.1, 2.5).Then, derive the quadratic interpolant through these two points and the point (1.2, 1.5). Show a graphdepicting this situation.

suppose that l and m are two lines. then the number of points in l intersect m is either 0, 1, or infinity