Question

What is the shortest possible perimeter for an arrangement with an area of 15 square feet

Answer 1

Answer: $4\sqrt{15}\ ft$

Step-by-step explanation:

Given

area of arrangement $A=15\ ft^2$

Suppose l and b the length and breadth of arrangement

Therefore perimeter is given by

$P=2(l+b)\quad \ldots(i)$

Area $A=lb=15$

$b=\frac{15}{l}$

Substitute the value of b in $(i)$

$P=2(l+\frac{15}{l})$

Differentiate P w.r.t l to get minimum perimeter

$\frac{dP}{dl}=2(1-\frac{15}{l^2})$

Substitute $\frac{dP}{dl}=0$

Thus

$1-\frac{15}{l^2}=0$

$l^2=15$

$l=\sqrt{15}$

Therefore $b=\frac{15}{\sqrt{15}}=\sqrt{15}$

Thus perimeter $P=2(\sqrt{15}+\sqrt{15})$

$P=4\sqrt{15}\ ft$