Question

A solid metallic cuboid of dimensions $10cm\times 5cm\times 3cm$ is melted and cast into a box, of thickness $0.5$ cm without lid. If the outer measurements of the base are $12cm\times 5cm$. find the height of the box.

Answer 1

First we need to find the Volume of the Cuboid, $V=10\times 5\times 3=150c{m}^3$

Now, Volume of Open box = Outer Volume - Inner Volume
Let $x$ be the height of the box

Outer Volume $=12\times 5\times x=60x$

Inner dimensions,$l=12-2\left(0.5\right)=12-1=11cm$, $b=5-2\left(0.5\right)=5-1=4cm,h=x-\left(0.5\right)$

Inner Volume $=11\times 4\times (x-0.5)=44(x-0.5)=44x-22$

Volume of box $=60x-(44x-22)=60x-44x+22=16x+22$

Since the cuboid is melted to form the box, Volume will be the same
$16x+22=150$
$=>16x=150-22$
$=>x=\frac{128}{16}$
$\therefore x=8cm$
Height is $8cm$