(a) (i) The height of each cone is equal to the diameter of its base - Leaving Cert Mathematics - Question 7 - 2017

To find the slant height $l$, we can use the Pythagorean theorem:

$l^2 = h^2 + r^2$

Substituting the values: $h = 4.5 \, m, \, r = 2.25 \, m$

Thus, $l^2 = (4.5)^2 + (2.25)^2 = 20.25 + 5.0625 = 25.3125$

Therefore, the slant height can be calculated as: $l = ext{sqrt}(25.3125) = 5.03 \, m.$

The curved surface area (CSA) of one cone is given by:

$CSA = \pi r l$

Substituting the values for one cone:

$CSA = \pi (2.25)(5.03) = 35.55 \, m²$

For 14 cones, the total CSA is:

$14 \times 35.55 = 497.77 \, m².$

Given that one litre of polish will cover 12-25 m², we first take the average coverage:

Average coverage = $\frac{12 + 25}{2} = 18.5 \, m²$

The total area to polish is 497.77 m², so:

$\text{Number of litres} = \frac{497.77}{18.5} ≈ 27$.
Rounded to the nearest litre, we need approximately 41 litres.

If one container holds 5 litres and costs A$110, to find the number of containers needed:

$\text{Number of containers} = \frac{41}{5} = 8.2 \approx 9 \, containers.$
This means 9 containers are needed, and thus the total cost in euro is:

9 \times 110 = A$990 \approx €673.

The length of the arc of the sector can be computed using:

$\text{Arc length} = \frac{\theta}{360°} \times 2\pi r$

For this question, substituting $r = 5.03$ m: $p = \frac{\theta}{360} \times 2 \pi (5.03) = 14.14 \, m.$

From the previous formula, locate θ:

\Rightarrow \theta = \frac{p \times 360}{2\pi r}$$ Using the values: $p = 14.14$, and $r = 5.03$, $$\theta = \frac{14.14 \times 360}{2\pi(5.03)} ≈ 161°.$$