Surface Areas and Volumes | Exercise 12.2

Question 4

A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand (see Fig. 12.16).

Solution

Understand the Question

A pen stand is created by carving $4$ identical conical depressions into a solid wooden cuboid.

To find the volume of wood remaining in the pen stand: $\text{Volume of wood} = \text{Volume of cuboid} - 4 \times (\text{Volume of one conical depression})$

$\text{Volume of cuboid} = l \times b \times h$
$\text{Volume of cone} = \dfrac{1}{3}\pi r^2 h$

Step 1 · Calculate the Volume of the Cuboid

Given dimensions of the cuboid: length $l = 15\text{ cm}$ , breadth $b = 10\text{ cm}$ , height $h = 3.5\text{ cm}$ . Diagram 1

\begin{aligned} V_{\text{cuboid}} &= l \times b \times h \\[0.6em] &= 15 \text{ cm} \times 10 \text{ cm} \times 3.5 \text{ cm} \\[0.6em] &= 525 \text{ cm}^3 \end{aligned}

Step 2 · Calculate the Volume of One Conical Depression

For each conical depression: radius $r = 0.5\text{ cm}$ , depth $h = 1.4\text{ cm}$ . Diagram 2

\begin{aligned} V_{\text{cone}} &= \dfrac{1}{3} \pi r^2 h \\[0.6em] &= \dfrac{1}{3} \times \dfrac{22}{7} \times (0.5 \text{ cm})^2 \times 1.4 \text{ cm} \\[0.6em] &= \dfrac{1}{3} \times \dfrac{22}{7} \times 0.25 \text{ cm}^2 \times 1.4 \text{ cm} \\[0.6em] &= \dfrac{1}{3} \times 22 \times 0.25 \times 0.2 \text{ cm}^3 \\[0.6em] &= \dfrac{1}{3} \times 1.1 \text{ cm}^3 \\[0.6em] &= \dfrac{1.1}{3} \text{ cm}^3 \approx 0.3667 \text{ cm}^3 \end{aligned}

Step 3 · Calculate the Total Volume of Four Conical Depressions

Diagram 3

Since there are $4$ identical depressions:

\begin{aligned} V_{\text{total depressions}} &= 4 \times V_{\text{cone}} \\[0.6em] &= 4 \times \dfrac{1.1}{3} \text{ cm}^3 \\[0.6em] &= \dfrac{4.4}{3} \text{ cm}^3 \\[0.6em] &= 1.4666... \text{ cm}^3 \approx 1.47 \text{ cm}^3 \end{aligned}

Step 4 · Calculate the Volume of Wood in the Stand

Subtract the total volume of depressions from the volume of the cuboid:

\begin{aligned} V_{\text{wood}} &= V_{\text{cuboid}} - V_{\text{total depressions}} \\[0.6em] &= 525 \text{ cm}^3 - 1.4666... \text{ cm}^3 \\[0.6em] &= 523.5333... \text{ cm}^3 \\[0.6em] &\approx 523.53 \text{ cm}^3 \end{aligned}

Answer

$523.53 \text{ cm}^3$

Common Mistakes

Adding Instead of Subtracting: Conical depressions are hollow cavities removed from the wood, so their volume must be subtracted, not added.
Missing the Count of Depressions: Forgetting to multiply the single cone's volume by $4$ .
Premature Rounding: Rounding the volume of one cone too early (e.g., to $0.37$ ) creates rounding inaccuracies in the final answer; keep exact fractional forms until the final step.

Question 4

More questions in Exercise 12.2