University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.1 - Functions and Their Graphs - Exercises - Page 11: 11

Answer

$1.\ \ \mathrm{Length\:of\:side\:as\:a\:function\:of\:diagonal:}\ \ a(d)=\frac{d\sqrt{3}}{3}$ $2.\ \ \mathrm{Volume\:as\:a\:function\:of\:diagonal:}\ \ A(d)=2d^2$ $3.\ \ \mathrm{Surface\:area\:as\:a\:function\:of\:diagonal:}\ \ V(d)=\frac{d^3}{3\sqrt{3}}$

Work Step by Step

$1.$ To find the length of the space diagonal, we need to find the length of the face diagonal of the cube. Consider the length of side $\ a$. By applying the pythagorean theorem to the triangle $\ \mathrm{HCD}$, we have: $(d_f)^2=a^2+a^2$ $\Rightarrow\ (d_f)^2=2a^2$ Now apply the pythagorean theorem to the triangle $\ \mathrm{FCH}\ $, to get the length of the space diagonal as: $d^2=(d_f)^2+a^2$ $\Rightarrow\ d^2=2a^2+a^2$ $\Rightarrow\ a^2=\frac{d^2}{3}$ $\Rightarrow\ a=\frac{d}{\sqrt{3}}$ $\Rightarrow\ a=\frac{d\sqrt{3}}{3}\ \ $ by rationalizing the denominator. So, the Length of side as a function of diagonal is, $\ \ a(d)=\frac{d\sqrt{3}}{3}$ $2.$ Formula for the surface area of a cube is $\ A=6a^2$. So, the surface area as a function of diagonal is, $\ \ A(d)=6(\frac{d^2}{3})=2d^2$ $3.$ Formula for the volume of a cube is $\ V=a^3$. So, the volume as a function of diagonal is, $\ \ V(d)=(\frac{d\sqrt{3}}{3})^3=\frac{d^3}{3\sqrt{3}}$
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