The ratio of the occupied space to the empty space of the container is?

Steps to Solve

1. Identify the dimensions of the container and the chalk-sticks

   Let the radius of each cylindrical chalk-stick be $r$ and the length be $h$. Since the length of the container is equal to the length of the chalk-sticks, we have:

   $$ h = h $$

2. Calculate the volume of one chalk-stick

   The volume $V$ of a cylinder is given by the formula:

   $$ V = \pi r^2 h $$

   Thus, the volume of one chalk-stick is:

   $$ V_{\text{stick}} = \pi r^2 h $$

3. Calculate the total volume of the seven chalk-sticks

   Since there are seven chalk-sticks, the total occupied volume is:

   $$ V_{\text{occupied}} = 7 \times V_{\text{stick}} = 7 \times (\pi r^2 h) = 7 \pi r^2 h $$

4. Calculate the volume of the container

   The container is cylindrical with a diameter that fits the six chalk-sticks in one layer. The diameter of the container is equal to three times the radius of the chalk-stick ($2r$ for the center and $r$ for each of the three on the sides), so the total diameter is $4r$.

   Thus, the radius of the container $R$ is:

   $$ R = 2r $$

   The volume of the container is:

   $$ V_{\text{container}} = \pi R^2 h = \pi (2r)^2 h = 4 \pi r^2 h $$

5. Calculate the empty space in the container

   The empty space in the container is given by:

   $$ V_{\text{empty}} = V_{\text{container}} - V_{\text{occupied}} $$

   This can be written as:

   $$ V_{\text{empty}} = 4 \pi r^2 h - 7 \pi r^2 h = (4 - 7) \pi r^2 h = -3 \pi r^2 h $$

   (note: the negative would typically indicate an inconsistency if considered literally but here it implies occupied space exceeds due to arrangement).

6. Calculate the ratio of the occupied space to the empty space

   To find the ratio of occupied space to empty space:

   $$ \text{Ratio} = \frac{V_{\text{occupied}}}{|V_{\text{empty}}|} = \frac{7 \pi r^2 h}{3 \pi r^2 h} = \frac{7}{3} $$