Page 385 - Excel for Scientists and Engineers: Numerical Methods

3 62 EXCEL: NUMERICAL METHODS

Problems

Data for, and answers to, the following problems are found in the folder "Ch. 15
(Random Numbers & Monte Carlo)" in the "Problems & Solutions" folder on the
CD.

1. Estimation of 7c. The equation of a circle is x2 + y2 = r2. Evaluate x by
determining the area of a circle of radius r circumscribed by a square of side
2r. x is the ratio of the area of the circle to that of the square. Generate a
pair of random numbers to use as the x and y coordinates. If the distance of
the point from the origin is less than or equal to r, it is within the circle.
Repeat this N times, evaluating Nc;, the number of points that fall within the
circle. The ratio Nc /N should be a reasonable estimate of 7r.

2. Male Children. A king wishes to increase the number of males in his
kingdom. He decrees that all women in his kingdom may have as many
children as they wish, as long as they are boys. As soon as a woman has a
female baby, she must stop bearing children. If this decree is followed, what
will be the ratio of boys to girls in the kingdom?
3. Traffic Model. Create a simple mode! of traffic patterns at a stoplight. Use
one row of a spreadsheet to represent a unit of time, say 5 seconds. Use a
random number to decide whether a car arrives at the intersection in a
particular time unit. Vary the traffic density (probability) and traffic light
timing; observe the effect on congestion at the stoplight.
4. Traveling Salesman. Given a number of cities and the costs of traveling
from any city to any other city, what is the cheapest round-trip route that
visits each city?

5. Choose Once. Using spreadsheet formulas only, create a list of unique
integers (e.g., 1-1 5) in random order.

6. Deck of Cards. Using spreadsheet formulas only, simulate the shuffling of a
deck of 52 cards.

7. Frequency of Occurrence of Digits. Create 1000 random numbers and
determine the frequency of occurrence of the numbers 0 through 9 in the first
digit.

8. Frequency of Occurrence of Digits 11. Create two columns, each
containing 1000 random numbers, RN1 and RN2. Determine the frequency
of occurrence in the first significant digit of the numbers 1 through 9 in the
product RN1 x RN2. Repeat for the product RN1 x RN2 x RN3.