Question

Chloe has enough sand to fill a sandbox with an area of 36 square units. She wants the outer edges of the sandbox to use as little material as possible.
Which dimensions will give Chloe the smallest perimeter?

Answer 1

Answer:

The dimensions of the box is 6 × 6 units.

Step-by-step explanation:

Chloe fill a sand box whose area is 36 square units.

Let length be x units and width be y units.

Area of box = 36

$xy=36\\y=\dfrac{36}{x}\ \ \ ...(i)$

Perimeter of box, $P=2(x+y)$

$P=2(x+\dfrac{36}{x})\ \ \ \ \ \ \ \ [\text{ From }(i)]$

Differentiate w.r to x

$\dfrac{dP}{dx}=2(1-\dfrac{36}{x^2})$

For critical point, $\dfrac{dP}{dx}=0$

$\therefore 2(1-\dfrac{36}{x^2})=0$

$\Rightarrow x=\pm6$

x can't be negative.

$\dfrac{d^2P}{dx^2}=\dfrac{144}{x^3}\\\dfrac{d^2P}{dx^2}|_{x=6}=\dfrac{144}{216}>0(\text{min})$

$P_{min}=24$ units

Hence, Length = 6 units and width = 6 units