0.2 Discrete structures problem solving  (Page 4/6)

From these equations we can easily obtain C = 223, S = 148, and T = 500. Thus the required percentages are 44.6%, 29.6%, and 25.8%, respectively.

All we had to do to solve this problem is to analyze relationships between the data and the unknowns, that is, nothing much beyond "understanding the problem".

Example 2

This is a problem which you can solve using similar known results.

Problem: Find the (length of) diagonal of a rectangular parallelepiped given its length, width and height.

Again let us try to solve this problem following the framework presented above.

Understanding the Problem: This is a "find" type problem. So we try to identify unknowns, data and conditions.

The unknown is the diagonal of a rectangular parallelepiped, and the data are its length, width and height. Again there are no explicitly stated conditions. But the unknown and data must all be a positive number.

Before proceeding to the next phase, let us make sure that we understand the terminologies. First a rectangular parallelepiped is a box with rectangular faces like a cube except that the faces are not necessarily a square but a rectangle as shown in Figure 1.

Thus   y2 = a2 + b2

is obtained from the second triangle, and

x2 = c2 + y2

is derived from the first triangle.

From these two equations, we can find that x is equal to the positive square root of   a2 + b2 + c2.

Example 3

This is a proof type problem and "proof by contradiction" is used.

Problem: Given that a, b, and c are odd integers, prove that equation ax2 + bx + c = 0 can not have a rational root.

Understanding the Problem: This is a "prove" type problem.

The hypothesis is that a, b, and c are odd integers, and the conclusion is that equation ax2 + bx + c = 0 can not have a rational root.

The hypothesis is straightforward. In the conclusion, "rational root" means a root, that is, the value of x that satisfies the equation, and that can be expressed as m/n, where m and n are integers. So the conclusion means that there is no number of the form m/n that satisfies the equation under the hypothesis.