Problem solving: Difficulty level: 700

[aishwaryav12]

Four identical cylinders are to be packed standing upright in the same direction into a rectangular shipping box with dimensions 3 x 12 x 4. What is the maximum possible volume of one of the cylinders?


A. 48Ï€
B. 27Ï€
C. 12Ï€
D. 9Ï€
E. 6.75Ï€

[deloitte247]

Given that width =3, Length =12, height =4
diameter of the article is at most 3 and the height is at most 4
Total volume per cylinder will be
$$\left(1.5\right)^2\cdot4\cdot\pi$$
$$=2.25\cdot4\cdot\pi$$
$$=9\pi$$
$$Answer\ =\ option\ D$$

[fskilnik@GMATH]

Four identical cylinders are to be packed standing upright in the same direction into a rectangular shipping box with dimensions 3 x 12 x 4. What is the maximum possible volume of one of the cylinders?
A. 48Ï€
B. 27Ï€
C. 12Ï€
D. 9Ï€
E. 6.75Ï€

\[? = {\left( {{V_{\,{\text{each}}\,\,{\text{cylinder}}}}} \right)_{\,\max }}\,\]
There are three possible ways of considering the rectangular shipping box:

Base: 3x12 and Height 4 -- we may put each upright cylinder in a base 3x3 (or 3/4 x 12, what is focus-worse) and height 4 , so that Vcylinder = [Ï€*(3/2)^2]*4 = 9Ï€
Base: 3x4 and Height 12 -- we may put each upright cylinder in a base 3x1 (or 3/4 x 4, what is focus-worse) and height 12 , so that Vcylinder = [Ï€*(1/2)^2]*12 = 3Ï€
Base 4x12 and Height 3 -- we may put each upright cylinder in a base 4x3 (or 1 x 12, what is focus-worse) and height 3 , so that Vcylinder = [Ï€*(3/2)^2]*3 = (27/4)Ï€
\[? = 9\pi \]
This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.