Question

A closed rectangular box with a square base is to be made so as to contain $1000$ cubic feet. The cost of the material per sq. foot for the bottom is $15$paisa, for the top $254$ paisa, and for the sides $20$ paisa. The labour charges for making the box are Rs. $3$. Find the dimensions of the box when the cost is minimum.

• A. $x=5,h=40$
• B. $x=20,h=2.5$
• C. $x=10,h=10$
• D. $x=2,h=125$

Answer 1

The correct option is C $x=10,h=10$

Let the dimension of the box with square base be $x,x,h$

Volume of the rectangular box $V=x^{2}h$

$x^{2}h=1000$

Area of top $=x^2$

Area of base $=x^2$

Area of faces $=4xh$

$E=15x^2+25x^2+20\left(4xh\right)+300$

$E=40x^2+80x\left(\frac{1000}{x^2}\right)+300$

$=40x^2+\frac{80\times 1000}{x}+300$

For maxima or minima,

$\frac{dE}{dx}=0$

$\therefore 80x-\frac{80\times 1000}{x^2}=0$

$\therefore x^3=1000$

$x=10$

$\frac{d^{2}E}{d{x}^2}=80+\frac{2\times 80\times 1000}{x^3}=+ive$

$\therefore$ Min. when $x=10$

$h=\frac{1000}{x^2}=10$

Hence the box should be a cube of edge 10 feet.