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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
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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
Step 2
The horizontal distance from A to B is 100 m. Use the triangle ABC to find the length of y.
Answer
We can use the sine rule in triangle ABC:
rac{y}{ ext{sin}(45°)} = rac{100}{ ext{sin}(105°)}.Rearranging gives:
y = 100 imes rac{ ext{sin}(45°)}{ ext{sin}(105°)}.Calculating the values:
y = 100 imes rac{0.7071}{0.9659} \ o y = 73.6 ext{ m} (to 1 decimal place).Step 3
Using your answer to Part (a)(ii) or otherwise, find the value of h and the value of x.
Answer
Using the triangle ABC again:
For height h:
rac{h}{ ext{sin}(30°)} = rac{y}{ ext{sin}(45°)} \ ext{Substituting } y = 73.6 ext{ m gives:} h = rac{73.6 imes ext{sin}(30°)}{ ext{sin}(45°)} = 51.8 ext{ m}.For length x:
Using triangle height:
rac{x}{ ext{sin}(30°)} = rac{100 ext{ m}}{ ext{sin}(105°)} \ o x = 36.6 ext{ m} (using known values).Step 4
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.
Answer
First calculate the cost of the cables:
Total length of cables = 2 * y = 2 * 73.6 = 147.2 m.
Cost of cables before VAT = €25 * 147.2 = €3680.
Cost after VAT = €3680 * 1.23 = €4526.4.
Cost of the mast:
Cost of mast before VAT = €580 * h = 580 * 51.8 = €30044.
Cost after VAT = €30044 * 1.23 = €36953.12.
Total cost = €4526.4 + €36953.12 = €41479.52.
Step 5
Find the area of the hexagon. Give your answer in km², correct to 2 decimal places.
Answer
The area A of a regular hexagon can be calculated using the formula:
A = rac{3 ext{sqrt}(3)}{2} s^2,where s is the side length. Here, s = 8 km:
A = rac{3 ext{sqrt}(3)}{2} * 8^2 \ = rac{3 ext{sqrt}(3)}{2} * 64 \ = 166.28 ext{ km}^2 ext{ (rounded to 2 decimal places)}.Step 6
Find this shaded area. Give your answer in km², correct to 1 decimal place.
Answer
The area of the circle that circumscribes the hexagon is given by:
A = rac{3 ext{sqrt}(3)}{2} * 8^2. \ ext{The area of the circle is } A = rac{3 imes ext{3.14}}{2} * 8^2 = 57.96 \ ext{Thus the shaded area is } 5.8 ext{ km}^2.