What is the volume of a pyramid with a regular pentagonal base of side 6 cm and a height of 10 cm?

The formula for the volume of a pyramid is given as $V = \frac{1}{3}Ah$, where $A$ is the area of the base and $h$ is the height.
From the image:

• The base is a regular pentagon with a side length ($s$) of $6 \text{ cm}$.
• The height ($h$) of the pyramid is $10 \text{ cm}$.

First, calculate the area of the regular pentagonal base ($A$). The formula for the area of a regular pentagon with side length $s$ is:

$$A = \frac{5s^2}{4\tan(\frac{180^\circ}{5})} = \frac{5s^2}{4\tan(36^\circ)}$$

Substitute $s = 6 \text{ cm}$:

$$A = \frac{5 \times (6 \text{ cm})^2}{4\tan(36^\circ)}$$ $$A = \frac{5 \times 36 \text{ cm}^2}{4\tan(36^\circ)}$$ $$A = \frac{180 \text{ cm}^2}{4\tan(36^\circ)}$$ $$A = \frac{45}{\tan(36^\circ)} \text{ cm}^2$$

Using $\tan(36^\circ) \approx 0.72654$:

$$A \approx \frac{45}{0.72654} \text{ cm}^2 \approx 61.937 \text{ cm}^2$$

Now, calculate the volume ($V$) of the pyramid:

$$V = \frac{1}{3}Ah$$

Substitute the calculated area and the given height:

$$V = \frac{1}{3} \times \left(\frac{45}{\tan(36^\circ)}\right) \text{ cm}^2 \times 10 \text{ cm}$$ $$V = \frac{150}{\tan(36^\circ)} \text{ cm}^3$$

Using the approximate value for $\tan(36^\circ)$:

$$V \approx \frac{150}{0.72654} \text{ cm}^3$$ $$V \approx 206.457 \text{ cm}^3$$

Rounding to two decimal places:

$$\boxed{206.46 \text{ cm}^3}$$

The formula for the volume of a sphere is given as $V = \frac{4}{3}\pi r^3$.
From the image:

• The radius ($r$) is $60 \text{ mm}$.
• Use $\pi = 3.14$.
• The answer should be given to the nearest cubic centimetre.

First, convert the radius from millimeters to centimeters:

$$r = 60 \text{ mm} = 6 \text{ cm}$$

Now, substitute the values into the volume formula:

$$V = \frac{4}{3} \times 3.14 \times (6 \text{ cm})^3$$ $$V = \frac{4}{3} \times 3.14 \times (6 \times 6 \times 6) \text{ cm}^3$$ $$V = \frac{4}{3} \times 3.14 \times 216 \text{ cm}^3$$ $$V = 4 \times 3.14 \times \frac{216}{3} \text{ cm}^3$$ $$V = 4 \times 3.14 \times 72 \text{ cm}^3$$ $$V = 12.56 \times 72 \text{ cm}^3$$ $$V = 904.32 \text{ cm}^3$$

Rounding to the nearest cubic centimetre:

$$\boxed{904 \text{ cm}^3}$$