Q) 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.

Solution:1. Occupied Space: The volume of one chalk-stick is \(\pi r^2 h\). For 7 chalk-sticks, the total volume occupied is: \[ V_{\text{occupied}} = 7 \pi r^2 h \] 2. Total Volume of the Container: The radius of the container is \(R = 3r\). Therefore, the volume of the container is: \[ V_{\text{container}} = \pi R^2 h = \pi (3r)^2 h = 9 \pi r^2 h \] 3. Empty Space: The empty space is: \[ V_{\text{empty}} = V_{\text{container}} - V_{\text{occupied}}\] \[= 9 \pi r^2 h - 7 \pi r^2 h = 2 \pi r^2 h \] 4. Ratio of Occupied Space to Empty Space: \[ \text{Ratio} = \frac{V_{\text{occupied}}}{V_{\text{empty}}} = \frac{7 \pi r^2 h}{2 \pi r^2 h} = \frac{7}{2} \] So, the ratio of the occupied space to the empty space is \(\frac{7}{2}\).