Sure, let's work through this step-by-step to determine the height of the pillar.
### Step-by-Step Solution:
1. Define the variables:
- Let [tex]\( h \)[/tex] be the height of the pillar.
- Let [tex]\( \theta \)[/tex] be the angle of elevation when viewed from 200 meters away.
- From a distance of 200 meters, the angle of elevation is [tex]\( \theta \)[/tex].
- From a distance of [tex]\( 200 - 125 = 75 \)[/tex] meters away, the angle of elevation is [tex]\( 2\theta \)[/tex].
2. Set up trigonometric equations:
We know that the tangent of an angle in a right triangle is the ratio of the opposite side (height of the pillar, [tex]\( h \)[/tex]) to the adjacent side (distance from the observer to the base of the pillar):
- From 200 meters away:
[tex]\[
\tan(\theta) = \frac{h}{200}
\][/tex]
- From 75 meters away:
[tex]\[
\tan(2\theta) = \frac{h}{75}
\][/tex]
3. Use the double angle formula for tangent:
Recall that:
[tex]\[
\tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan(\theta)^2}
\][/tex]
Substitute [tex]\(\tan(\theta) = \frac{h}{200}\)[/tex] into the double angle formula:
[tex]\[
\tan(2\theta) = \frac{2 \left(\frac{h}{200}\right)}{1 - \left(\frac{h}{200}\right)^2}
\][/tex]
4. Equate the expressions for [tex]\(\tan(2\theta)\)[/tex]:
[tex]\[
\frac{h}{75} = \frac{2 \left(\frac{h}{200}\right)}{1 - \left(\frac{h}{200}\right)^2}
\][/tex]
5. Simplify the equation:
First, simplify the right side of the equation:
[tex]\[
\frac{h}{75} = \frac{\left(\frac{2h}{200}\right)}{1 - \left(\frac{h^2}{200^2}\right)} = \frac{\left(\frac{h}{100}\right)}{1 - \left(\frac{h^2}{40000}\right)}
\][/tex]
Next:
[tex]\[
\frac{h}{75} = \frac{h / 100}{1 - h^2 / 40000}
\][/tex]
6. Cross-multiply to get rid of the fractions:
[tex]\[
\frac{h}{75} \left(1 - \frac{h^2}{40000}\right) = \frac{h}{100}
\][/tex]
7. Distribute and simplify:
[tex]\[
\frac{h}{75} - \frac{h^3}{75 \cdot 40000} = \frac{h}{100}
\][/tex]
Simplifying further:
[tex]\[
\frac{h}{75} - \frac{h^3}{3000000} = \frac{h}{100}
\][/tex]
8. Isolate [tex]\( h \)[/tex]:
Multiply everything by 3000000 to clear the fractions:
[tex]\[
40000h - h^3 = 30000h
\][/tex]
Combine like terms:
[tex]\[
10000h = h^3
\][/tex]
9. Solve the equation:
[tex]\[
h^3 = 10000h
\][/tex]
[tex]\[
h^2 = 10000
\][/tex]
[tex]\[
h = \sqrt{10000}
\][/tex]
10. Calculate [tex]\( h \)[/tex]:
[tex]\[
h = 100 \text{ meters}
\][/tex]
### Conclusion:
The height of the pillar is [tex]\( 100 \)[/tex] meters.
