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

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Understand the Problem

The question is asking for the ratio of the occupied space to the empty space in a cylindrical container holding seven identical cylindrical chalk-sticks. It provides the context that the length of the container is equal to the length of the chalk-sticks.

Answer

The ratio of the occupied space to the empty space is $\frac{7}{3}$.
Answer for screen readers

The ratio of the occupied space to the empty space of the container is $\frac{7}{3}$.

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:

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

The ratio of the occupied space to the empty space of the container is $\frac{7}{3}$.

More Information

This problem illustrates the concept of volumes of cylinders and how to compute ratios between occupied and empty spaces. The arrangement of the chalk-sticks and the geometry involved are crucial for understanding the spatial distribution.

Tips

  • Failing to account for the arrangement of the objects, leading to incorrect volume calculations.
  • Not realizing that the radius for the container needs to be calculated based on the configuration of the chalk-sticks.
  • Misinterpreting the occupied and empty space calculations which can lead to confusion with signs.
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