Points A, B and the lighthouse are related by angles of elevation.
- The distance of the lighthouse from A is 97.4 m
- The height of the lighthouse is 79.9 m
From the complete question (see attachment), we have:
[tex]\mathbf{\angle A = 164 - 90}[/tex]
[tex]\mathbf{\angle A = 74}[/tex]
[tex]\mathbf{\angle B = 270 - 218}[/tex]
[tex]\mathbf{\angle B = 52}[/tex]
[tex]\mathbf{\angle L = 180 - 74 - 52}[/tex]
[tex]\mathbf{\angle L = 54}[/tex]
(a) The distance between the lighthouse and A
This is calculated using the following sine ratio
[tex]\mathbf{\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}}[/tex]
So, we have:
[tex]\mathbf{\frac{AL}{sin B} = \frac{100}{sin L}}[/tex]
This gives
[tex]\mathbf{\frac{AL}{sin 52} = \frac{100}{sin 54}}[/tex]
Make AL the subject
[tex]\mathbf{AL = \frac{100}{sin 54} \times sin 52}[/tex]
[tex]\mathbf{AL = 97.4}[/tex]
The distance of the lighthouse from A is 97.4 m
(b) The height of the lighthouse
Considering triangle AHL, we have:
[tex]\mathbf{\angle H = 90 + 27 = 117}[/tex]
[tex]\mathbf{\angle A = 74 - 27 = 47}[/tex]
[tex]\mathbf{AL = 97.4}[/tex]
The height is then calculated using the following sine ratio
[tex]\mathbf{\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}}[/tex]
So, we have:
[tex]\mathbf{\frac{AL}{sin H} = \frac{HL}{sin A}}[/tex]
Make HL the subject
[tex]\mathbf{HL = \frac{AL \times sin A}{sin H}}[/tex]
This gives
[tex]\mathbf{HL = \frac{97.4 \times sin 47}{sin 117}}[/tex]
[tex]\mathbf{HL = 79.9}}[/tex]
Hence, the height of the lighthouse is 79.9 m
Read more about elevation and depressions at:
https://brainly.com/question/21137209
