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Page 18: Practice Problems

Exercise 24

Step 1

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$$
textbf{underline{textit{Solution}}}
$$

Step 2

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enumerate[bfseries (a)]
item
The plot is shown in the following textbf{graph}, where the mass is plotted on $y$-axis and the volume on $x$-axis

Exercise scan

Step 3

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enumerate[bfseries (b)]

item The resulting curve that describes the relation between the mass and the volume of the nuggets, is a straight line passing through the origin.

Step 4

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enumerate[bfseries (c)]
item The relation between the mass and the volume of the nuggets is a directly linear proportional relation, in the sense that as the volume of the nuggets increases the mass of the nuggets also increase.
[ m propto V]

Step 5

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enumerate[bfseries (d)]
item The slope of the line that best fits all the points is calculated using excel, to be 19.372, where we can calculate the slope after plotting the points on the graph and drawing the best fit line using the following equation
[ m = dfrac{y_2 – y_1}{x_2 – x_1}]
Where, we can calculate the slope from any two points having a coordinate ($x_1,y_1$) and ($x_2,y_2$) on the fitting line.\

As, the $y$-axis is the mass (gm), and the $x$-axis is the volume (cm$^3$) thus the slope would have a unit of gm/cm$^3$.\

textbf{underline{textit{note:}}} we do not calculate the slope from the point given in the table, as those points may have a greater error, the fitting line is the line that describes the relation between both variables with the least error so we calculate the slope from the fitting line”.\

Step 6

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enumerate[bfseries (e)]
item Since the straight line is passing through origin and have a slope of $m=19.372$, thus the equation of the straight line is
[ m=19.372 ~ V]

Step 7

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enumerate[bfseries (f)]
item The slope of this lines yields the density of the pure gold nuggets which is approximately 19.4 gm/cm$^3$.

Result

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enumerate[bfseries (a)]
item See graph.
item Straight line passing through the origin.
item $m propto V$
item The slope of the straight line is 19.372 with a unit of gm/cm$^3$.
item The equation of the straight line is $m=19.372 V$.
item The density of the pure gold nuggets is approximately 19.4 gm/cm$^3$

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Page 18: Practice Problems

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