Question

Barun needs an open box of capacity $864m^3$. Actually where he lives, the rates of paints are soaring high so he wants to minimize the surface area of the box keeping the capacity of the box same as required. What is the base area and height of such a box?

• A. $36m^2,24m$
• B. $216m^2,4m$
• C. $144m^2,6m$
• D. None of these

Answer 1

The correct option is C $144m^2,6m$

Shortcut: Very quickly check the options. If all the options have values.

Alternatively:
The required capacity of box is $864m^3$. Let the length of the base l and height of the box is 'h', then
$864=l^{2}h\Rightarrow h=\frac{864}{l^2}$
Now, surface area of the box $A=l^2+4lh=l^2+\frac{4l\times 864}{l^2}=l^2+\frac{3456}{l}$
Differentiating w.r.t l, we get
$\frac{dA}{dl}=2l-\frac{3456}{l^2}$
For the minimum area $=\frac{dA}{dl}=0$
$\begin{array}{cc}\therefore & 2l=\frac{3456}{l^2}\\ \Rightarrow & l^3=1728\\ \Rightarrow & l=12\\ \therefore & Basearea=(12{)}^2=144\\ and& Height=\frac{864}{l^2}=6m\end{array}$