Can you determine p experimentally?

You are conducting an experiment to see if you can experimentally determine the value of p. The main idea of the experiment is to graph random points onto a circle inscribed in a square. and then determine the ratio of the number of circle points to the number of points in one of the quarter squares.

Performing the experiment:

Graph a circle with center at (0,0) and radius of 10 on graph paper. Draw square ABCD with A(0,0), B(10,0),C(10,10) and D(10,0). You will be locating points on this graph and recording if the point falls in the circle or in the square.

You will work with a partner to determine your random points. Type the following on your graphing calculator:

  1. Reset your random number generator by storing your birthday into rand. For example, 23 STO rand (rand in located in the MATH menu, under PRB).
  2. Get a random number between 0 and 10 by going to the MATH key, choosing PRB, then selecting choice 5: randInt(0,10)
  3. Press Enter to get a new random number. You do not need to retype the command.

You must complete this part during class:

Create a x/y table for at least 100 points. Use your calculator's random numbers to fill in the x column. (Press ENTER without retyping the command to get a long list.) Use your partner's list to fill in your y column. You will not have the same points as your partner. (The x and y coordinates will be switched.) If you want, you can use your calculator to get the y coordinates as well.

Plot your points on your graph. Record if each point falls in the circle or outside. Points inside the circle satisfy the following inequality: x2 + y2 <= 100. Use this to check any points that look close to the edge of the circle.

Count the number of points that are in the square (including edges and corners.) Some points will be counted twice: once as a circle point and once as a small square point.

Calculate the ratio of the number of circle points to the number of points in the square.

Writing Tips

Your report should use the following format. Use the bold words as titles for each section.

A. Description of the experiment

B. Hypothesis: what do you think will result and why? Be specific, include the calculation of the ratio of the area of the part of the circle to the area of the square. Write this ratio in two ways: 1) in terms of P and 2) as a decimal.

C. Data organized in tables and/or graphs

D. Data Analysis: how do your results compare to your hypothesis? Be specific, include the calculation of the ratio of circle points to the number of points in the square. Write this ratio as a decimal to compare it to the theorectical ratio calculated in B.

E. Conclusion: how are your results related to the value of p? How much error was there in your result? How could the experiment be improved?