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: 9

Answer

$\mathrm{Perimeter:}\ \ P(a)=3a$ $\mathrm{Area:}\ \ A(a)=\frac{\sqrt{3}}{4}a^2$

Work Step by Step

In an equilateral triange, all three sides are equal $\ \ i.e,\ \ a=b=c $ We know that the perimeter is the sum of all the sides. Perimeter $=P=a+b+c$ $=3a\ \ $ since all the sides are of equal length. Write this formula as a function of $\ a\ $ as: $ P(a)=3a$ If we cut the equilateral triangle into half, we will get a right triangle with its adjacent of length $\ \frac{a}{2}\ $, opposite side $\ h\ $, and hypotenuse $\ a$. We know that the area of a triangle is half the product of its base and height, $\ \ A=\frac{a\times h}{2}$. We need the value of $\ h\ $. Using Pythagorean's Theorem : $a^2=(\frac{a}{2})^2+h^2$ $\Rightarrow\ h^2=a^2-\frac{a^2}{4}$ $\Rightarrow\ h^2=\frac{3a^2}{4}$ $\Rightarrow\ h=\frac{\sqrt{3}}{2}a$ So, the area will be written as: $A=\frac{a\times\frac{\sqrt{3}}{2}a}{2}$ $\Rightarrow\ A=\frac{\sqrt{3}}{4}a^2$ Area as a function of $\ a\ $ is: $A(a)=\frac{\sqrt{3}}{4}a^2$
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