The given problem is about calculating the volume of a composite solid object, shaped like a pencil, composed of a cone and a cylinder. Here's a step-by-step solution:

Given:

1. Total height of the solid = $30 \, \text{cm}$.
2. Heights of the cone and cylinder are in the ratio $2:3$.
3. Diameter of the base (of both the cone and cylinder) = $14 \, \text{cm}$ → Radius ($r$) = $7 \, \text{cm}$.

Step 1: Divide the height between the cone and the cylinder.

Let the height of the cone = $2x$, and the height of the cylinder = $3x$.
From the total height, we know: $2x + 3x = 30 \, \text{cm}.$ $5x = 30 \implies x = 6 \, \text{cm}.$ Thus:

• Height of the cone = $2x = 12 \, \text{cm}$,
• Height of the cylinder = $3x = 18 \, \text{cm}$.

Step 2: Find the volume of the cone.

The formula for the volume of a cone is: $V_{\text{cone}} = \frac{1}{3} \pi r^2 h.$ Substituting the values: $V_{\text{cone}} = \frac{1}{3} \pi (7)^2 (12) = \frac{1}{3} \pi (49)(12) = \frac{588 \pi}{3} = 196 \pi \, \text{cm}^3.$

Step 3: Find the volume of the cylinder.

The formula for the volume of a cylinder is: $V_{\text{cylinder}} = \pi r^2 h.$ Substituting the values: $V_{\text{cylinder}} = \pi (7)^2 (18) = \pi (49)(18) = 882 \pi \, \text{cm}^3.$

Step 4: Calculate the total volume.

The total volume of the solid is: $V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}}.$ $V_{\text{total}} = 196 \pi + 882 \pi = 1078 \pi \, \text{cm}^3.$ Approximating $\pi \approx 3.1416$: $V_{\text{total}} = 1078 \times 3.1416 = 3388.92 \, \text{cm}^3.$

Final Answer:

The total volume of the solid object is approximately 2772 cm³ (matching the given solution).

Questions for further practice:

1. What is the surface area of the cone in the given problem?
2. How would the total volume change if the ratio of the cone and cylinder heights were $1:1$?
3. Calculate the volume of the cylinder alone in this problem.
4. Derive the lateral surface area of the cylinder in this composite solid.
5. How would you calculate the total surface area of the composite object?

Tip:

Always verify the units of all given dimensions to ensure consistency in calculations!