a) The diagram shows a circle inscribed in a square - Leaving Cert Mathematics - Question Question 1 - 2012

First, we need to calculate the area of the circle. The formula for the area of a circle is:

$$A = \pi r^2 .$$

Substituting in our radius:

$$A_{circle} = \pi (2)^2 = 4\pi \approx 12.6 \text{ cm}².$$

Now, we can calculate the shaded area, which is the area of the square minus the area of the circle:

$$\text{Shaded Area} = 16 - 4\pi \approx 16 - 12.6 = 3.4 \text{ cm}².$$

To find the height of the candle, we first calculate the volume of the wax in the candle, which comprises a cylinder and a cone.

The volume of the cylinder is given by the formula:

$$V_{cylinder} = \pi r^2 h .$$

With a radius of ( r = \frac{3}{2} = 1.5 \text{ cm} ) and height ( h = 8 \text{ cm} ), we have:

$$V_{cylinder} = \pi (1.5)^2 (8) = \pi (2.25)(8) = 18\pi \text{ cm}^3.$$

The volume of the cone is:

$$V_{cone} = \frac{1}{3} \pi r^2 h .$$

Using the same radius and a height of 4 cm (as the cone is on top), we find:

$$V_{cone} = \frac{1}{3} \pi (1.5)^2 (4) = \frac{1}{3} \pi (2.25)(4) = 3\pi .$$

Thus, we calculate the total height of the candle as:

$$\text{Height of candle} = 8 + 4 = 12 \text{ cm} .$$

Since nine candles fit into a rectangular box and the base of the box is square, we determine the dimensions of the box.

Given that each candle has a base diameter of 3 cm, we can fit three candles wide, leading to the box's base dimensions being:

$$\text{Base Dimensions} = 3 \times 3 = 9 \text{ cm} .$$

The height of the box is determined by adding the height of the candle:

$$\text{Height} = 12 \text{ cm} .$$

Finally, the volume of the box is given by:

$$\text{Volume} = \text{Base Area} \times \text{Height} = 9^2 \times 12 = 81 \times 12 = 972 \text{ cm}^3 .$$