Unless stated otherwise, take . 1. A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of .

Understand the Question The solid is composed of two parts: a cone standing on top of a hemisphere, both sharing the same circular base. Given: Radius of the hemisphere and cone, Height of the cone, The total volume of the solid is the sum of the volumes of both shapes: The final answer must be kept in terms of . Step 1 · Calculate Volume of the Cone Given: Radius of cone, Height of cone, Volume of the cone: Step 2 · Calculate Volume of the Hemisphere Given: Radius of hemisphere, Volume of the hemisphere: Step 3 · Find the Total Volume of the Solid The total volume is the sum of both volumes: Answer > Common Mistakes Substituting : The question specifically requests the answer in terms of , so do not replace with or . Full Sphere Formula: Using instead of for the hemisphere. Incorrect Units: Writing square units () instead of cubic units () for volume.

Surface Areas and Volumes | Exercise 12.2

Question 1

Unless stated otherwise, take $\pi = \dfrac{22}{7}$ .

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$ .

Solution

Understand the Question

The solid is composed of two parts: a cone standing on top of a hemisphere, both sharing the same circular base.
Given:
- Radius of the hemisphere and cone, $r = 1\text{ cm}$
- Height of the cone, $h = r = 1\text{ cm}$
The total volume of the solid is the sum of the volumes of both shapes: $\text{Total Volume} = V_{\text{cone}} + V_{\text{hemisphere}} = \dfrac{1}{3}\pi r^2 h + \dfrac{2}{3}\pi r^3$
The final answer must be kept in terms of $\pi$ .

Step 1 · Calculate Volume of the Cone

Given:

Radius of cone, $r = 1\text{ cm}$
Height of cone, $h = 1\text{ cm}$

Volume of the cone:

\begin{aligned} V_{\text{cone}} &= \dfrac{1}{3}\pi r^2 h \\[0.6em] &= \dfrac{1}{3} \times \pi \times (1\text{ cm})^2 \times (1\text{ cm}) \\[0.6em] &= \dfrac{1}{3} \times \pi \times 1\text{ cm}^2 \times 1\text{ cm} \\[0.6em] &= \dfrac{1}{3}\pi\text{ cm}^3 \end{aligned}

Step 2 · Calculate Volume of the Hemisphere

Given:

Radius of hemisphere, $r = 1\text{ cm}$

Volume of the hemisphere:

\begin{aligned} V_{\text{hemisphere}} &= \dfrac{2}{3}\pi r^3 \\[0.6em] &= \dfrac{2}{3} \times \pi \times (1\text{ cm})^3 \\[0.6em] &= \dfrac{2}{3} \times \pi \times 1\text{ cm}^3 \\[0.6em] &= \dfrac{2}{3}\pi\text{ cm}^3 \end{aligned}

Step 3 · Find the Total Volume of the Solid

The total volume is the sum of both volumes: $\text{Total Volume} = V_{\text{cone}} + V_{\text{hemisphere}}$

\begin{aligned} V_{\text{solid}} &= \dfrac{1}{3}\pi\text{ cm}^3 + \dfrac{2}{3}\pi\text{ cm}^3 \\[0.6em] &= \left(\dfrac{1}{3} + \dfrac{2}{3}\right)\pi\text{ cm}^3 \\[0.6em] &= \left(\dfrac{1+2}{3}\right)\pi\text{ cm}^3 \\[0.6em] &= \dfrac{3}{3}\pi\text{ cm}^3 \\[0.6em] &= \pi\text{ cm}^3 \end{aligned}

Answer

$\pi\text{ cm}^3$

Common Mistakes

Substituting $\pi$ : The question specifically requests the answer in terms of $\pi$ , so do not replace $\pi$ with $\dfrac{22}{7}$ or $3.14$ .
Full Sphere Formula: Using $\dfrac{4}{3}\pi r^3$ instead of $\dfrac{2}{3}\pi r^3$ for the hemisphere.
Incorrect Units: Writing square units ( $\text{cm}^2$ ) instead of cubic units ( $\text{cm}^3$ ) for volume.

Question 1

More questions in Exercise 12.2