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
[tex]xy=36\\y=\dfrac{36}{x}\ \ \ ...(i)[/tex]
Perimeter of box, [tex]P=2(x+y)[/tex]
[tex]P=2(x+\dfrac{36}{x})\ \ \ \ \ \ \ \ [\text{ From }(i)][/tex]
Differentiate w.r to x
[tex]\dfrac{dP}{dx}=2(1-\dfrac{36}{x^2})[/tex]
For critical point, [tex]\dfrac{dP}{dx}=0[/tex]
[tex]\therefore 2(1-\dfrac{36}{x^2})=0[/tex]
[tex]\Rightarrow x=\pm6[/tex]
x can't be negative.
[tex]\dfrac{d^2P}{dx^2}=\dfrac{144}{x^3}\\\dfrac{d^2P}{dx^2}|_{x=6}=\dfrac{144}{216}>0(\text{min})[/tex]
[tex]P_{min}=24[/tex] units
Hence, Length = 6 units and width = 6 units
