GATE Civil 2024 Set 2 | GA | Question: 7

Question

Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container. The figure below shows the arrangement of the chalk-sticks inside the cylinder.

The length of the container is equal to the length of the chalk-sticks. The ratio of the occupied space to the empty space of the container is

• A. $5 / 2$
• B. $7 / 2$
• C. $9 / 2$
• D. $3$

Answer 1

$\text{Ratio of the occupied space to the empty space of the container} = \dfrac{\text{Total volume occupied by chalk-sticks in cylindrical container}}{\text{Total volume remained in cylindrical container after tightly fitted chalk-sticks}}$

$= \dfrac{7 * \text{Total volume of one chalk-stick}}{\text{Total volume of cylindrical container} - 7 \times \text{Total volume of one chalk-stick}}$

$= \dfrac{ 7 \times \pi r^{2} h}{\pi R^{2} h - 7 \times \pi r^{2} h}$

$= \dfrac{ 7 \pi r^{2} h}{\pi (3r)^{2} h - 7 \pi r^{2} h} \quad (R = 3r \, [\text{From the fig. given in question}])$

$= \dfrac{ 7 \pi r^{2} h}{9 \pi r^{2} h - 7 \pi r^{2} h}$

$= \dfrac{ 7 \pi r^{2} h}{2 \pi r^{2} h}$

$= \dfrac{7}{2}$

Answer 2

b