Find the height of a tower on an inclined hill using angles of elevation

Expert intro: This is a classic right-triangle-on-a-slope setup. The hill is already tilted, so the 54° angle of elevation is measured from the horizontal line at the observer, not from the hillside itself. That detail matters because the hill’s 16° incline changes the geometry a lot.

Detailed walkthrough

Let the tower height be $h$ feet.

1) Split the geometry into horizontal and vertical parts

The observer is 95 ft down the hill, and the hill is inclined $16^\circ$ above the horizontal.

So relative to the point at the tower’s base:

• horizontal distance = $95\cos 16^\circ$
• vertical change = $95\sin 16^\circ$ downward

If the tower stands vertically on top of the hill, then the top of the tower is $h$ feet above the hilltop.

2) Use the angle of elevation

From the observer, the angle of elevation to the top is $54^\circ$, so

$$\tan 54^\circ=\frac{h+95\sin 16^\circ}{95\cos 16^\circ}$$

Solve for $h$:

$$h=95\cos 16^\circ\tan 54^\circ-95\sin 16^\circ$$

3) Compute the value

$$95\cos 16^\circ\tan 54^\circ\approx 95(0.9599)(1.3764)\approx 125.4$$ $$95\sin 16^\circ\approx 95(0.2756)\approx 26.2$$

So

$$h\approx 125.4-26.2=99.2$$

Answer

$\boxed{99.2\text{ ft (approximately)}}$

💡 Pitfall guide

A common mistake is to treat the 95 ft as if it were horizontal. It is not — it lies along the hill. Another easy slip is using $\sin 54^\circ$ instead of $\tan 54^\circ$. Here the tangent ratio fits because we are comparing vertical rise to horizontal run from the observer.

🔄 Real-world variant

If the 95 ft had been measured horizontally instead of along the slope, the setup would change to

$$\tan 54^\circ=\frac{h+95\tan 16^\circ}{95}$$

That gives a different height. So when the problem says “down the hill,” always check whether the distance is along the incline or on level ground.

🔍 Related terms

angle of elevation, inclined plane, right triangle trigonometry