Question

Dimension of box :Height =$x$ inch,Length =$24-2x$ inch, Breadth= $9-2x$

What is the maximum volume of the box?

• A. $200$ cubic inch
• B. $400$ cubic inch
• C. $100$ cubic inch
• D. None of the above

Answer 1

The correct option is A $200$ cubic inch

So the dimension of that rectangular are $\left(\right(24-2x)\times (9-2x)\times x)$

And volume of that rectangular box = $(24-2x)(9-2x)\left(x\right)$

for this to be maximum $\frac{dV}{dx}=0$ for some $x<9$

so,

$(volume{)}^{\prime }=(-2\left)\right(9-2x\left)\right(x)+(24-2x\left)\right(-2\left)\right(x)+(24-2x\left)\right(9-2x)$

$0=(-2)(-33x+4x^2)+(216+4x^2-66x)$

$12x^2=216$

$x=2inch$

Maximum volume = $200$ cubic inches