Question

A metal box with a square base and vertical height is to contain 1024 $c{m}^2$. The material for the top and the bottom costs Rs.5/ $c{m}^2$ and the material for the sides costs Rs. 2.50/$c{m}^2$. Find the least cost of the box.

Answer 1

Surface area of box $=1024c{m}^2$

S.A$=a^2\times 2+(a\times h)\times 4$

$1024=2a^2+4ah$

Cost of square bases$=$Rs. $5/c{m}^2$

$\therefore$ For $2a^2=$Rs. $2a^2\times 5$

$=$Rs. $10a^2$

Cost of rectangular faces$=$Rs. $50/c{m}^2$

$\therefore$ For $4ah=$Rs. $2.50\times 4ah$

$=$Rs. $10ah$

Least cost$=$Rs. $10a^2+$Rs. $10ah$.