Help understanding a distance problem

[kari82]

This is a review problem, I have the solution but I cannot understand the way my professor solved the problem. Can someone be kind enough to help me understand? Thanks!

A student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. He cuts out various shapes in various sizes, calculates their areas, measures their masses, and prepares the graph of Figure I4 (see attachment). Consider the fourth experimental point from the top. How far is it from the best-fit straight line?

(a) Express your answer as a difference in vertical axis coordinate.
(b) Express your answer as a difference in horizontal-axis coordinate.
(c) Express both of the answers to parts (a) and (b) as a percentage.
(d) Calculate the slope of the line.
(e) State what the graph demonstrates, referring to the shape of the graph and the results of parts (c) and (d).
(f) Describe whether this result should be expected theoretically. Describe the physical meaning of the slope.



(a) We read from the graph a vertical separation of 0.3 spaces = 0.015 g.
(b) Horizontally, 0.6 spaces = 30 cm2.
(c) Because the graph line goes through the origin, the same percentage describes the
vertical and the horizontal scatter: 30 cm2/380 cm2 = 8%.
(d) Choose a grid point on the line far from the origin: slope = 0.31 g/600 cm2 = 0.000 52
g/cm2 = (0.000 52 g/cm2)(10 000 cm2/1 m2) = 5.2 g/m2 .
(e) For any and all shapes cut from this copy paper, the mass of the cutout is proportional
to its area. The proportionality constant is 5.2 g/m2 ± 8%, where the uncertainty is
estimated.
(f) This result should be expected if the paper has thickness and density that are uniform
within the experimental uncertainty. The slope is the areal density of the paper, its mass per unit area.

 

[anathema]

This means that the paper has a consistent thickness and density throughout, which is expected in a stack of copy paper. The graph demonstrates a linear relationship between the mass and the area of the cutouts, with a proportionality constant of 5.2 g/m2. This result is expected theoretically because the paper should have a consistent thickness and density throughout, leading to a proportional relationship between mass and area. The slope of the line represents the areal density of the paper, which is an important physical characteristic of the paper.