A vertical mobile phone mast, [DC], of height h m, is secured with two cables: [AC] of length x m, and [BC] of length y m, as shown in the diagram - Leaving Cert Mathematics - Question 7 - 2020

In triangle ABC, using the angle at B (45°) and the opposite side (y), we can apply the sine rule:

It is known that:

y = 100 \times \frac{\sin 45°}{\sin 30°}

Calculating:

• (\sin 45° = \frac{\sqrt{2}}{2})
• (\sin 30° = \frac{1}{2}) Substituting:

y = 100 \times \frac{\frac{\sqrt{2}}{2}}{\frac{1}{2}} = 100 \times \sqrt{2} \approx 100 \times 1.4142 = 141.42\ m\n So, to one decimal place, y = 141.4 m.

Using the previously found value of y (141.4 m) in triangle ACD, we can find h:

Given (\tan(30°) = \frac{h}{100}) (h = 100 \times \tan(30°)) Using (\tan(30°) = \frac{1}{\sqrt{3}} \approx 0.5774):

h = 100 \times 0.5774 \approx 57.74 m, to 1 decimal place h = 57.7 m.

Next, for x in triangle ACB:

(\sin(30°) = \frac{y}{x}) (x = \frac{y}{\sin(30°)}) Substituting:

x = \frac{141.4}{0.5} = 282.8 m.

Thus, to 1 decimal place, x = 282.8 m.

First, compute the cost of the cables:

Cables cost = 25 \times 282.8 = €7070.

Mast cost = 580. Total cost before VAT: = 7070 + 580 = €7650.

Now, calculate the VAT: VAT = 23% of €7650 = 0.23 \times 7650 = €1749.50.

Finally, the total cost including VAT: Total cost = 7650 + 1749.50 = €9399.50.

The area A of a regular hexagon can be calculated using the formula:

$$A = \frac{3\sqrt{3}}{2} s^2$$

where s is the length of a side. Here, s = 8 km.

Calculating the area:

$$A = \frac{3\sqrt{3}}{2} (8)^2 = \frac{3\sqrt{3}}{2} \times 64 = 96\sqrt{3} \approx 166.28 \ km².$$

Thus, the area is 166.28 km², correct to 2 decimal places.

The area of the circle inscribed in the hexagon can be calculated using the formula for the area of a circle:

$$Area = \pi r^2$$

To find r (the radius): For a regular hexagon, the radius r is given by: (r = \frac{s}{\sqrt{3}}), where s is the side length. Here, r = \frac{8}{\sqrt{3}} \approx 4.6188 km.

Thus, the area:

$$Area = \pi (4.6188)^2 \approx 66.94 km².$$

Therefore, the shaded area is approximately 66.9 km², correct to 1 decimal place.