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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.

Next a diagonal of a rectangular parallelepiped is the line that connects its two vertices (corner points) that are not on the same plane. It is shown in Figure 2.

Devising a Solution Plan: Here we first try to find relevant facts. Relevant facts often involve the same or similar words or concepts. Since the unknown is a diagonal, we look for facts concerning diagonal. Note that drawing figures here is quite helpful. One of the facts that immediately comes to our mind in this problem is Pythagoras' theorem. It has to do with right triangles and is shown in Figure 3.

Let us try to see whether or not this theorem helps. To use this theorem, we need a right triangle involving a diagonal of a parallelepiped. As we can see in Figure 4, there is a right triangle with a diagonal x as its hypotenuse.

However, the triangle here involves two unknowns: x and y. Since x is what we are looking for, we need to find the value of y. To find y, we note another right triangle shown in Figure 5.

Applying Pythagoras' theorem again, we can obtain the value of y.

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.

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Read also:

OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1

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