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. The angle of elevation to the top of the mast from A is 30° and from B is 45°. (a) (i) Explain why ∠LBCA is 105°. (ii) The horizontal distance from A to B is 100 m. Use the triangle ABC to find the length of y. Give your answer correct to 1 decimal place. (iii) Using your answer to Part (a)(ii) or otherwise, find the value of h and the value of x. Give your answers correct to 1 decimal place. (b) The two cables to secure the mast costs €25 per metre. The mast itself costs €580 per metre. VAT at 23% is then added in each case. Calculate the total cost of the cables and mast after VAT is included. (c) (i) The mast can provide a strong signal for an area in the shape of a regular hexagon of side 8 km, as shown in the diagram. Find the area of the hexagon. Give your answer in km², correct to 2 decimal places. (ii) A circle which touches all vertices of the hexagon can show areas where the signal is weak. One of these areas is shaded in the diagram. Find this shaded area. Give your answer in km², correct to 1 decimal place.

Leaving Cert Mathematics Geometry: A vertical mobile phone mast, [D

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

Question 7

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.png

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. The angle of elev... show full transcript

Worked Solution & Example Answer: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

Step 1

Explain why ∠LBCA is 105°.

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Answer

In triangle ABC, the sum of angles in a triangle equals 180°. Given that ∠A is 30° and ∠B is 45°, we can calculate ∠C as follows:

∠C = 180° - (∠A + ∠B) = 180° - (30° + 45°) = 180° - 75° = 105°.

Thus, the angle ∠LBCA is 105°.

Step 2

The length of y.

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Answer

Using the sine rule in triangle ABC:

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

Solving for y:

y = 100 \times \frac{\sin 30°}{\sin 105°} = 100 \times \frac{0.5}{0.9659} ≈ 51.8 m.

Thus, the length of y is approximately 51.8 m.

Step 3

Find the value of h and the value of x.

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Answer

Using triangle ABC again:

For h:

Using the sine rule:

\frac{h}{\sin 45°} = \frac{y}{\sin 30°} = \frac{51.8}{0.5}.

This gives:

h = \frac{51.8 \times \sin 45°}{\sin 30°} = \frac{51.8 \times 0.7071}{0.5} ≈ 73.6 m.

For x:

Using the sine rule:

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

Substituting h:

x = \frac{h \times \sin 30°}{\sin 45°} = \frac{73.6 \times 0.5}{0.7071} ≈ 52.2 m.

Thus, the height h is approximately 73.6 m and x is approximately 52.2 m.

Step 4

Calculate the total cost of the cables and mast after VAT is included.

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Answer

Cost of cables = 2 \times 25 \times x m\n Cost of mast = 580 Euros. Total cost before VAT:

Total = (2 \times 25and \times 73.2) + 580 = 29,584.19 Euros.

Including VAT of 23%:

TotalCost = TotalCost \times 1.23 = 29,584.19 \times 1.23 = 35,172.10 Euros.

Step 5

Find the area of the hexagon.

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Answer

Area of a regular hexagon with side length s is given by:

Area = \frac{3\sqrt{3}}{2} s^2.

For s = 8 km:

Area = \frac{3\sqrt{3}}{2} (8)^2 = 166.28 km².

Thus, the area is approximately 166.28 km².

Step 6

Find this shaded area.

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Answer

The area of the circle inscribed in the hexagon is given by:

Area = \frac{3\sqrt{3}}{2}}s^2 - \text{Area of hexagon}.

This calculation results in an area of approximately 5.8 km².

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

Worked Solution & Example Answer: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

Explain why ∠LBCA is 105°.

The length of y.

Find the value of h and the value of x.

Calculate the total cost of the cables and mast after VAT is included.

Find the area of the hexagon.

Find this shaded area.

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