Four Application Math Problems

The following are four mathematics problems. Do the best you can on them. You are not necessarily expected to solve all these problems perfectly. I am mainly interested in how you explore problems and present your reasoning. Also, if you do not enjoy doing these problems, then you probably would not enjoy this summer program. This style of problem is one of the types of problem you would see during the program, but the subject matter is not necessarily the same.

(1) Suppose two lines L and R intersect at a 60° angle.

Starting at a point A0 on line L, take steps of equal length, alternating between the two

lines, never stepping back to the spot just used. [Note: All the steps have the same length.]

Label the successive points of the path A0, A1, A2, A3, . . .

After some number of steps, do you return to your starting point A0? Does the answer depend on the location of the starting point? Justify your answers.

[NOTE: You might first investigate this situation with a compass and ruler!

One key: Remember that an isosceles triangle (a triangle with two equal sides) has equal base angles.]

(2) Estimate the volume of your body in terms of number of jellybeans.

(3) The letters a1, a2, a3, a4, a5, a6, a7 represent seven positive whole numbers. The

letters b1, b2, b3, b4, b5, b6, b7 represent the same numbers but in a different order. Will

the value of the product:

(a1 – b1)(a2 – b2)(a3 – b3)(a4 – b4)(a5 – b5)(a6 – b6)(a7 – b7)

always be an even number? Explain your conclusion.

(4) Call a convex pentagon with one set of parallel sides “obedient.” Call a pentagon with two sets of parallel sides “very obedient.”

(a) Can the longest side of a “very obedient” pentagon not be in one of the parallel pairs? Either way, can you prove or argue your belief?

(b) Can you come up with an alternate way to describe a “very obedient” pentagon? What else can you say about them?