To solve this problem, let's start by identifying and setting up the variables:
1. Let [tex]\( h \)[/tex] be the height of the shorter post.
2. Therefore, the height of the taller post is [tex]\( 2h \)[/tex].
3. The distance between the posts is 120 meters. So, from the midpoint, the horizontal distance to each post is [tex]\( 60 \)[/tex] meters.
Using the information provided:
- From the midpoint of the 120-meter distance, the observer sees the tops of the posts at complementary angles.
- Complementary angles mean that the sum of the two angles is [tex]\( 90^\circ \)[/tex] or [tex]\( \frac{\pi}{2} \)[/tex] radians.
Let's denote the angle of elevation to the top of the shorter post as [tex]\( \theta \)[/tex].
Then, the angle of elevation to the top of the taller post is [tex]\( 90^\circ - \theta \)[/tex].
Using trigonometric relationships, we have:
1. For the shorter post:
[tex]\[ \tan(\theta) = \frac{h}{60} \][/tex]
2. For the taller post and using complementary angles [tex]\( 90^\circ - \theta \)[/tex]:
[tex]\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{\frac{h}{60}} = \frac{60}{h} \][/tex]
But at the same time, we know:
[tex]\[ \tan(90^\circ - \theta) = \frac{\text{height of taller post}}{\text{horizontal distance}} \][/tex]
[tex]\[ \text{height of taller post} = 2h \][/tex]
[tex]\[ \tan(90^\circ - \theta) = \frac{2h}{60} = \frac{h}{30} \][/tex]
Setting the two expressions for [tex]\( \tan(90^\circ - \theta) \)[/tex] equal, we get:
[tex]\[ \frac{60}{h} = \frac{h}{30} \][/tex]
Solving for [tex]\( h \)[/tex]:
[tex]\[ \frac{60}{h} = \frac{h}{30} \][/tex]
[tex]\[ 60 \times 30 = h \times h \][/tex]
[tex]\[ 1800 = h^2 \][/tex]
[tex]\[ h = \sqrt{1800} \][/tex]
[tex]\[ h = \sqrt{36 \cdot 50} \][/tex]
[tex]\[ h = \sqrt{36} \cdot \sqrt{50} \][/tex]
[tex]\[ h = 6 \cdot \sqrt{50} \][/tex]
[tex]\[ h = 6 \cdot \sqrt{25 \cdot 2} \][/tex]
[tex]\[ h = 6 \cdot 5 \cdot \sqrt{2} \][/tex]
[tex]\[ h = 30\sqrt{2} \][/tex]
Therefore, the height of the shorter post is [tex]\( 30\sqrt{2} \)[/tex] meters.
