A penstand made up of wood in the shape of a cuboid with four conical depressions to hold pens the dimensions of the cuboid are 15cm by 10cm by 3.5 cm. Find the radius of the each of the depressions 

To solve this problem, we need to calculate the radius of each conical depression in the cuboid pen stand.

**Step 1: Understanding the Shape and Volume**

- The pen stand is a cuboid with dimensions \( 15 \, \text{cm} \times 10 \, \text{cm} \times 3.5 \, \text{cm} \).
- There are four conical depressions to hold pens. We need to determine the radius of each of these depressions.

### Assumptions:
- The depressions are perfect cones.
- The depth of each conical depression will be equal to the height of the cuboid (which is 3.5 cm).
- The total volume of the cuboid minus the volume of the conical depressions is not explicitly mentioned, but for simplicity, we will assume the cones do not overlap and their entire volume lies within the cuboid.

**Step 2: Volume of the Cones**

The volume \( V \) of a cone is given by the formula:

\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]

Where:
- \( r \) is the radius of the base of the cone.
- \( h \) is the height of the cone, which is 3.5 cm (since the depth of the depression is equal to the height of the cuboid).

**Step 3: Volume of the Cuboid**

The volume \( V_{\text{cuboid}} \) of the cuboid is given by:

\[
V_{\text{cuboid}} = \text{length} \times \text{width} \times \text{height}
\]

Substitute the dimensions:

\[
V_{\text{cuboid}} = 15 \, \text{cm} \times 10 \, \text{cm} \times 3.5 \, \text{cm} = 525 \, \text{cm}^3
\]

**Step 4: Total Volume of the Four Conical Depressions**

Let the volume of each cone be \( V_{\text{cone}} \), and since there are four cones:

\[
4V_{\text{cone}} = 4 \times \frac{1}{3} \pi r^2 h
\]

\[
4V_{\text{cone}} = \frac{4}{3} \pi r^2 \times 3.5
\]

Simplifying:

\[
4V_{\text{cone}} = \frac{14}{3} \pi r^2
\]

**Step 5: Subtracting the Volume of the Cones from the Volume of the Cuboid**

We can assume that the volume of the cuboid minus the volume of the cones should still account for most of the solid wood. To proceed, we need either the remaining solid volume or the volume of the conical depressions, or additional details. 

Let me know if you'd like to continue with different assumptions or provide further details! 

Would you like more clarification or assistance with other steps? Here are some related problems:

1. How do you calculate the surface area of the cuboid after the depressions?
2. How would the result change if the depth of the cones was halved?
3. How would the problem be solved if the cone's height was different from the cuboid's height?
4. What is the volume of a cuboid if only half of its height was used?
5. How does the formula for a cone's volume differ from that of a cylinder?

**Tip**: When solving for unknown dimensions, always clarify which volumes or surfaces you need to subtract or include to avoid confusion!

A penstand made up of wood in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15cm by 10cm by 3.5 cm. Find the radius of each of the depressions.

Learn how to find the radius of four conical depressions in a wooden cuboid penstand with dimensions 15 cm by 10 cm by 3.5 cm. This geometry problem explores volume calculation for cuboids and cones. Ideal for students in Grades 8-10.

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