Solving Tower Height with Complementary Angles

This problem involves finding the height of a tower using the concept of complementary angles of elevation from two different points on the ground.

Understanding Complementary Angles of Elevation

Complementary angles add up to $90^\circ$. If the angle of elevation from one point is $\alpha$, the angle from the second point (which is closer or further away) will be $90^\circ - \alpha$.

Let:

• $h$ be the height of the tower.
• $d_1 = 25$ m be the distance from the first point.
• $d_2 = 16$ m be the distance from the second point.
• $\alpha$ be the angle of elevation from the first point.
• $90^\circ - \alpha$ be the angle of elevation from the second point.

Applying Trigonometry

Using the tangent function, we can relate the height ($h$), distances, and angles:

• From the first point: $\tan(\alpha) = \frac{h}{d_1} = \frac{h}{25}$
• From the second point: $\tan(90^\circ - \alpha) = \frac{h}{d_2} = \frac{h}{16}$

Using Trigonometric Identities

We know the identity $\tan(90^\circ - \theta) = \cot(\theta)$. Also, $\cot(\theta) = \frac{1}{\tan(\theta)}$.

Applying this to our second equation:

$\cot(\alpha) = \frac{h}{16}$

Since $\cot(\alpha) = \frac{1}{\tan(\alpha)}$, we can write:

$\frac{1}{\tan(\alpha)} = \frac{h}{16}$

Calculating the Tower Height

Now substitute $\tan(\alpha) = \frac{h}{25}$ into the equation:

$\frac{1}{(\frac{h}{25})} = \frac{h}{16}$

Simplify the left side:

$\frac{25}{h} = \frac{h}{16}$

Cross-multiply to solve for $h^2$:

$h^2 = 25 \times 16$

$h^2 = 400$

Take the square root to find the height $h$:

$h = \sqrt{400}$

$h = 20$ m

Conclusion

The height of the tower is 20 m.