2. Directions 1.With a method similar to that of the previous lab, you will be using a Rutherford analysis to uncover a value hidden in your data. 2.Begin by placing a sheet of carbon paper over your lab sheet. 3.Using a marble, drop it randomly across the carbon paper at least 200 times. 4.Count the number of dots just within the squares on your paper. Try to estimate the dot’s position based on its center. 5.Count the number of dots within both the squares and the circles on your paper. Try to estimate the dot’s position based on its center. 6.Divide the number of dots within circles by the number of dots within the squares. (dots inside the squares are also inside the circles). 7.Does the resulting number seem familiar to you? Multiply it by two, and you may be surprised. 8.If you aren’t sure what the actual value should be, read the attached rationale. Dots within squares:____________ Dots within circles :_____________ Total Number of Drops:__________ Result:_______________________ Percent Error:_________________

3. Rationale By now, you might have figured what our mystery number was, but let’s see if we can work it out mathematically. If your drops really were random, the probability that a dot would land inside a circle or a square or both – and thus the number of dots within each – is directly proportional to areas of the two figures. We know that the area of a circle is πr 2, and the area of a square is s 2, where s is the length of one side of the square. To find the length of a side, we divide the square into four equally sized quadrants, creating four smaller squares. In the diagram below, r is the radius of the circle, and x is length of one side one of the small squares. By the Pythagorean Theorem: x 2 + x 2 = r 2 2x 2 = r 2 x = r√½ Therefore, the s of our larger square is 2x, or 2r√½. Putting this value into the equation for the area (A) of our larger gives us: A = s 2 A = (2r√½) 2 A = (4/2)r 2 A = 2r 2 Divide the area of our circle (πr 2 ) by the area of our square (2r 2 ), and you get the theoretical value you should have gotten for our lab. Now calculate your percent error, and you’re done! r x x