Question

A Toblerone bar comes in packaging that forms a triangular prism with equilateral triangles at both ends. If the volume of the package is 20cm³, determine the dimensions of the package that give a minimum surface area. V = 1/2bhl, S = bh + 3bl 

Answer 1

Given-

A Toblerone bar comes in packaging that forms a triangular prism with equilateral triangles at both ends.

So, its length and breadth is equal means $b=l$

$V=20cm^3$ $V=\dfrac{1}{2bhl}$

A.T.Q $S=bh+3bl$

$\Rightarrow$ $S=\dfrac{1}{2Vl}+3l^2$ $($ $V=\dfrac{1}{2bhl},$ $b=l)$

$\Rightarrow$ differentiate the above equation with respect to $l$

$\Rightarrow$ $\dfrac{dS}{dl}=\dfrac{-l^2}{2V}+6l^3$

put this equation equal to zero for minimum surface area

$\Rightarrow$ $l=\dfrac{1}{12V} =0.0046 cm$

which is equal to $b=0.0046 cm$

then $h=1181.47cm$ $(h=\dfrac{1}{2Vbl})$