1. Let the width of the rectangle be a, and its height be b.
a*b=216, so b=216/a
2. the perimeter of the rectangle is P=2a+2b=2a+2(216/a)=2a+432/a
since a varies, and P also varies accordingly, we can write P as a function in variable a as follows:
[tex]P(a)=2a+432a ^{-1} [/tex]
3. Taking the derivative we find the critical points where we have the largest or smallest values the function may take:
[tex]P'(a)=2-432a ^{-2}=2- \frac{432}{ a^{2} } [/tex]
[tex]2- \frac{432}{ a^{2} }=0[/tex]
[tex]\frac{432}{ a^{2} }=2[/tex]
[tex]\frac{216}{ a^{2} }=1[/tex]
[tex]a= \sqrt{216}=14.7 [/tex] or [tex]a= \sqrt{216}=-14.7 [/tex]
4. Since [tex]P(a)=2a+432a ^{-1}=2a+ \frac{432}{a} [/tex] would be negative with the negative root, the largest value of P is when a=14.7
5. b=216/a=216/14.7=14.7
Answer: [14.7, 14.7] (or [square root 216, square root 216])
