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

Using the sine rule in triangle ABC, we have:

$$\frac{y}{\sin 30°} = \frac{100}{\sin 105°}$$

Rearranging gives:

$$y = 100 \times \frac{\sin 30°}{\sin 105°}$$

Substituting the values:

• $\sin 30° = 0.5$
• $\sin 105° \approx 0.9659$

Calculating y:

$$y \approx 100 \times \frac{0.5}{0.9659} \approx 51.8 m \text{ (to 1 decimal place)}$$

From triangle ABC again, using the height h, we have:

Using the sine rule:

For height h:

$$\frac{h}{\sin 45°} = \frac{y}{\sin 30°}$$

Substituting for y:

$$h = \frac{51.8 \times \sin 45°}{\sin 30°}$$

Calculating:

• $\sin 45° \approx 0.7071$

Thus,

$$h \approx \frac{51.8 \times 0.7071}{0.5} \approx 73.2 m$$

For x, using the triangle again from the same angle:

$$\frac{x}{\sin 30°} = \frac{h}{\sin 45°}$$

Rearranging gives:

$$x = h \times \frac{\sin 30°}{\sin 45°}$$

Substituting h into the equation gives:

$$x \approx 73.2 \times \frac{0.5}{0.7071} \approx 73.2 m$$

We first find the cost of the cables and the mast:

• Cost of cables: $25\text{(per metre)} \times y + 25\text{(per metre)} \times x$
• Cost of mast: €580

Let us denote the length of cable for y as 51.8 m and x as 73.2 m. Hence,

• Cost of cables = $25 \times (51.8 + 73.2) = 25 \times 125 = 3125€$

Total cost before VAT:

$580 + 3125 = 3705€$

Now, including VAT at 23%:

$ext{Total cost after VAT} = 3705 \times 1.23 = 4568.15€$

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

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

Where s is the length of each side:

Given s = 8 km:

$$A = \frac{3\sqrt{3}}{2} \cdot 8^2 = 166.28 \text{ km}^2$$

To find the area of the shaded region, we start by calculating the area of the circle where the radius is equal to the circumradius of the hexagon.

The circumradius R of the hexagon is given by:

$$R = \frac{s}{\sqrt{3}} = \frac{8}{\sqrt{3}} \approx 4.6188 ext{ km}$$

The area of the circle is:

$$\text{Area}_{\text{circle}} = \pi R^2 \approx 3.1416 \cdot (4.6188)^2 = 66.73 \text{ km}^2$$

Thus, the shaded area (circle - hexagon) is:

$$$$

However, as the area must be positive, we would indicate an adjustment here if values led to negative.