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: 10

Answer

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

Work Step by Step

$1.$ Let's say, $\ a\ $ is the side length of the square $\square ABCD$. If we cut the square into half, we will get two isosceles triangles. Consider the triangle $\triangle\mathrm{BCD}$. In the considered triangle, let's suppose $\ d\ $ represents the length of diagonal. To represent the length of a side as a function of a diagonal, we need to apply the Pythagorean Theorem as: $d^2=a^2+a^2$ $\Rightarrow\ 2a^2=d^2$ $\Rightarrow\ a=\frac{d}{\sqrt{2}}$ $\Rightarrow\ a=\frac{d\sqrt{2}}{2}\ \ $ by rationalizing the denominator. So, length of the side as a function of the diagonal is, $\ \ a(d)=\frac{d\sqrt{2}}{2}$ $2.$ We know that the area of a square is equal to the square of its length of a side. $A=a^2$ $\Rightarrow\ A=(\frac{d\sqrt{2}}{2})^2$ $\Rightarrow\ A=\frac{d^2\cdot 2}{4}$ $\Rightarrow\ A=\frac{d^2}{2}$ So, area as a function of diagonal is, $A(d)=\frac{d^2}{2}$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.