Cones is a sculpture in the National Gallery of Australia - Leaving Cert Mathematics - Question 7 - 2017

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

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

Substituting the known values:

$l^2 = (4.5)^2 + (2.25)^2$

Calculating:

$l^2 = 20.25 + 5.0625 = 25.3125$

Thus,

$l = ext{sqrt}(25.3125) \approx 5.03 ext{ m}$

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

$CSA = ext{π} r l$

Substituting for one cone:

$CSA = ext{π} (2.25)(5.03)$

Calculating:

$CSA \approx 35.358$

Therefore, for 14 cones:

$ext{Total CSA} = 14 \times 35.358 \approx 495.003 \text{ m}^2$

To find the number of litres needed:

$\text{No. of litres} = \frac{\text{Total Area}}{\text{Coverage per litre}}\n\text{No. of litres} = \frac{495.003}{12.25} \approx 40.41\n$

Rounding to the nearest litre, we need 41 litres.

First, calculate the number of containers needed:

$\text{No. of containers} = \frac{41}{5} = 8.2 \approx 9 ext{ containers}\n$

Next, calculate the total cost:

$\text{Cost} = 9 \times 110 = 990 \text{ AUD}\n$

Now convert to euro:

$\text{Cost in euro} = 990 \times 0.68 = 673.2\n$

Thus, rounding to the nearest euro, the cost is €673.

The arc length p is given by the formula:

$p = 2\pi r\left(\frac{\theta}{360}\right)\n$

Substituting r and solving for p:

$p = 2\pi(2.25) \left(\frac{180}{360}\right) \approx 14.14 ext{ m}$

Using the arc length formula:

$p = \frac{\theta}{360} \cdot 2\pi r \n\Rightarrow \theta = \frac{p \cdot 360}{2\pi r}\n$

Substituting the values:

$\theta = \frac{14.14 \cdot 360}{2 \cdot \pi \cdot 5.03} \approx 161.48 \text{ degrees} \approx 161\degree$