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.
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:
sin30°y=sin105°100.
Solving for y:
y=100×sin105°sin30°=100×0.96590.5≈51.8m.
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:
sin45°h=sin30°y=0.551.8.
This gives:
h=sin30°51.8×sin45°=0.551.8×0.7071≈73.6m.
For x:
Using the sine rule:
sin30°x=sin45°h.
Substituting h:
x=sin45°h×sin30°=0.707173.6×0.5≈52.2m.
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: