1. **State the problem:** We need to find the height $P$ of the lamp using the given measurements and the angle of elevation. 2. **Identify the known values:** - Height of the theodolite: $1.6$ m - Distance from the theodolite to the lamp: $4$ m - Angle of elevation to the top of the lamp: $37^\circ$ 3. **Set up the problem using trigonometry:** The angle of elevation forms a right triangle where: - The adjacent side is the horizontal distance $4$ m - The opposite side is the height difference between the lamp and the theodolite, which is $P - 1.6$ 4. **Use the tangent function:** $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ Substitute the known values: $$\tan(37^\circ) = \frac{P - 1.6}{4}$$ 5. **Solve for $P$:** Multiply both sides by 4: $$4 \times \tan(37^\circ) = P - 1.6$$ Add $1.6$ to both sides: $$P = 4 \times \tan(37^\circ) + 1.6$$ 6. **Calculate the value:** Using $\tan(37^\circ) \approx 0.7536$, $$P = 4 \times 0.7536 + 1.6 = 3.0144 + 1.6 = 4.6144$$ 7. **Final answer:** The height of the lamp $P$ is approximately **4.61 meters**.