Question

Graphical Analysis and Techniques ​

Procedure

The goal of this exercise is for you to determine the relationship and constant of proportionality between the radius and area of a circle. You may already know what this relationship is, but here you will attempt to “prove” it to yourself. You'll be provided the diameters of several circles, from which you can find the respective radii. The areas of the circles will be found by an independent method. If we then plot a graph of area vs radius for these circles, hopefully the shape of the curve generated will suggest what the relationship is a allow you to “zero in on it” just like in the above example.

We will be using some data collected from circles of varying size cut out from rigid sheets of paper. If we first determine the area of the rectangular sheets of paper and measure their mass, we can compute the density of the paper. Thus the area of the cut out circles can be determined by measuring their mass and using the same density value.

Let's define the two-dimensional (or surface) density as: D = m/A

where m is the the mass, and A is the area it covers. Since a cut out of this same paper will have the same density of the entire sheet, we can solve for the area by using the same density and measured mass. Thus we have: A = m/D.

Below is a set of data collected for two sheets of paper used to generate the circles we'll use. That is followed by the dimensions of the cut out circles.

Table 1: Measurements of Paper Sheets

	Mass (g)	Length (cm)	Width (cm)	Area (cm2)	Density (g/cm2)
Sheet 1	9.198	27.93	21.63
Sheet 2	9.104	28.01	21.62

Average =

Table 2: Measurements of Paper Circles

Diameters (cm)	Mass (g)	Area (cm2)	Radius (cm)	Radius2 (cm2)
4.88	0.308
6.19	0.481
7.09	0.624
7.89	0.768
9.15	1.012
10.35	1.271
11.75	1.667
15.63	2.889

1. Complete the area and density values in Table 1. Be sure to provide one sample calculation of each here and remember to limit the digits appropriately. The area of a rectangle is length times width. Also, fill in the average density at the bottom of the table.

2. Using the average density found for Table 1, use the masses of the circles in Table 2 to determine their respective area. Please provide one sample calculation here. Also compute the radii values from the diameters in Table 2.

Answer 1

Solution:

Table 1: Measurements of Paper Sheets

Area of a reactangle = length x width

Therefore, for the first sheet, Area = 27.93 x 21.63 = 604.13 cm²

For the second sheet, Area = 28.01 x 21.62 = 605.58 cm²

Density of the paper = mass/ area

For first sheet, density = (9.198 / 604.13) = 0.0152 g/cm²

For second sheet, density = (9.104 / 605.58) = 0.0150 g/cm²

	Mass (g)	Length (cm)	Width (cm)	Area (cm2)	Density (g/cm2)
Sheet 1	9.198	27.93	21.63	604.13	0.0152
Sheet 2	9.104	28.01	21.62	605.58	0.0150

Therefore, average density of the paper,D = 0.0151 g/cm²

Table 2: Measurements of Paper Circles

Using the average density D = 0.0151 g/cm² of the paper and the mass of the paper circles we get the area of the paper by the formula, Area = mass/ average density

For the first set of data, mass =0.308 gm for which the area = ( 0.308/ 0.0151) = 20.397cm²

For which the radius can be calculated from the diameter value given = 4.88 cm

Therefore the radius = 2.44 cm

radius² = (2.44)² = 5.954 cm²

Diameters (cm)	Mass (g)	Area (cm2)	Radius (cm)	Radius2 (cm2)
4.88	0.308	20.397	2.440	5.954
6.19	0.481	31.854	3.095	9.579
7.09	0.624	41.325	3.545	12.567
7.89	0.768	50.861	3.945	15.563
9.15	1.012	67.019	4.575	20.930
10.35	1.271	84.172	5.175	26.780
11.75	1.667	110.397	5.875	34.516
15.63	2.889	191.325	7.815	61.074