Book edition 9th Edition

Author(s) Raymond A. Serway, John W. Jewett

Pages 1624 pages

ISBN 9781133947271

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Short Answer

Figure P1.10 shows a frustum of a cone. Match each of theexpressions(a) $π (r_{1} {+r}_{2}) {[h^{2} + {(r_{2} - r_{1})}^{2}]}^{1 / 2}$ , $π (r_{1} {+r}_{2}) {[h^{2} + {(r_{2} - r_{1})}^{2}]}^{1 / 2}$ (b) and(c) $π h ({r_{1}}^{2} + r_{1} r_{2} + {r_{2}}^{2}) / 3$ with the quantity it describes: (d) the total circumference of the flat circular faces, (e) the volume, or (f) the area of the curved surface.

The matches are given by:

(a) and (f)

(b) and (c)

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: A concept and analysis of the given data

The analysis of a relationship between different physical quantities by using the units of measurements and dimensions is called dimensional analysis. It is used to examine the correctness of an equation.

The formula of a circumference is $C = 2 π r,$ and its dimension is $L .$

The formula of a volume is $V = π r^{2} \frac{h}{3},$ and its dimension is $L^{3} .$

The formula of an area is $A = π r (r + \sqrt{h^{2} + r^{2}}),$ and its dimension is $L^{2} .$

Step 2: Find the dimensions of equation (a)

The given equation is $π (r_{1} {+r}_{2}) {[h^{2} + {(r_{2} - r_{1})}^{2}]}^{1 / 2} .$

In the above formula, $π$ has no dimension.

The dimensionsof $r_{1}$ , $r_{2},$ and $h$ is $L$

Replace for $r_{1}$ , $r_{2},$ and $h$ in the above equation to get its dimension.

$\begin{matrix} D i m e n s i o n = (L+L) {[L^{2} + {(L - L)}^{2}]}^{1 / 2} \\ = L {(L^{2})}^{1 / 2} \\ = L (L) \\ = L^{2} . \end{matrix}$

Hence, the dimension of equation (a) is equal to the dimension of the area of the curved surface.

Step 3: Find the dimensions of equation (b)

The equation given is $2 π (r_{1} {+r}_{2}) .$

In the above formula, $π$ has no dimension.

The dimensionsof $r_{1}$ , $r_{2},$ and $h$ is $L .$

Replace $L .$ for $r_{1}$ , and $r_{2}$ in the above equation to get its dimension.

$\begin{matrix} Dimenstion = (L + L) \\ = L . \end{matrix}$

Hence, the dimension of equation (b) is equal to the dimension of the circumference of the flat circular base.

Step 4: Find the dimensions of equation (c)

$π h ({r_{1}}^{2} + r_{1} r_{2} + {r_{2}}^{2}) / 3$ Consider the given equation as shown below.

In the above formula, $π$ has no dimension.

The dimensionsof $r_{1}$ , $r_{2},$ and $h$ is $L$

Replace for , $r_{1}$ $r_{2},$ and $h$ in the above equation to get its dimension.

$\begin{matrix} D i m e n s i o n = L (L^{2} + L L + L^{2}) \\ = L (L^{2}) \\ = L^{3} . \end{matrix}$

Hence, the dimension of the equation (c) is equal to the dimension of the volume.

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Physics For Scientists & Engineers

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Short Answer

Step by Step Solution

Step 1: A concept and analysis of the given data

Step 2: Find the dimensions of equation (a)

Step 3: Find the dimensions of equation (b)

Step 4: Find the dimensions of equation (c)

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Physics For Scientists & Engineers

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Short Answer

Step by Step Solution

Step 1: A concept and analysis of the given data

Step 2: Find the dimensions of equation (a)

Step 3: Find the dimensions of equation (b)

Step 4: Find the dimensions of equation (c)

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