Answer:
The shortest possible perimeter is: [tex]4\sqrt{15}[/tex]
Step-by-step explanation:
Given
[tex]A = 15[/tex] --- area
Required
Find the shortest possible perimeter
Area is calculated as:
[tex]A=l * b[/tex]
This gives:
[tex]l * b = 15[/tex]
Make l the subject
[tex]l=\frac{15}{b}[/tex]
Perimeter is calculated as:
[tex]P =2(l + b)[/tex]
Substitute [tex]l=\frac{15}{b}[/tex]
[tex]P =2(\frac{15}{b} + b})[/tex]
Rewrite as:
[tex]P =2(15b^{-1} + b)[/tex]
Differentiate and minimize;
[tex]P' =2(-15b^{-2} + 1)[/tex]
Minimize by equating P' to 0
[tex]2(-15b^{-2} + 1) = 0[/tex]
Divide through by 2
[tex]-15b^{-2} + 1 = 0[/tex]
[tex]-15b^{-2} = -1[/tex]
Divide through by -1
[tex]15b^{-2} = 1[/tex]
Rewrite as:
[tex]\frac{15}{b^2} = 1[/tex]
Solve for b^2
[tex]b^2 = \frac{15}{1}[/tex]
[tex]b^2 = 15[/tex]
Solve for b
[tex]b = \sqrt{15[/tex]
Recall that: [tex]P =2(\frac{15}{b} + b})[/tex]
[tex]P = 2 * (\frac{15}{\sqrt{15}} + \sqrt{15})[/tex]
[tex]P = 2 * (\sqrt{15} + \sqrt{15})[/tex]
[tex]P = 2 * (2\sqrt{15})[/tex]
[tex]P = 4\sqrt{15}[/tex]
The shortest possible perimeter is: [tex]4\sqrt{15}[/tex]
