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 giv…

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asked Mar 3, 2021 121k views

2 Answers

Qualle · Answer 1 · 2021-03-09T22:14:03+0000

Answer:

The dimensions of the box is 6 × 6 units.

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

$xy=36\\y=(36)/(x)\ \ \ ...(i)$

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

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

Differentiate w.r to x

(dP)/(dx)=2(1-(36)/(x^2))

For critical point,
(dP)/(dx)=0

$\therefore 2(1-(36)/(x^2))=0$

$\Rightarrow x=\pm6$

x can't be negative.

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

P_(min)=24 units

Hence, Length = 6 units and width = 6 units