Two buildings are 66 m apart. From the top of the shorter building, the angle of depression to the foot of the taller building is 51 degrees, and the angle of elevation to the top of the taller buildi

Let $h_1$ be the height of the first building, $h_1 = 66 \text{ m}$.
Let $h_2$ be the height of the taller building.
Let $x$ be the horizontal distance between the two buildings.
From the top of the first building, the angle of depression to the foot of the taller building is $51^\circ$. This forms a right-angled triangle where the opposite side is $h_1$ and the adjacent side is $x$.

$$\tan(51^\circ) = \frac{h_1}{x}$$

From the top of the first building, the angle of elevation to the top of the taller building is $34^\circ$. This forms another right-angled triangle where the opposite side is $h_2 - h_1$ and the adjacent side is $x$.

$$\tan(34^\circ) = \frac{h_2 - h_1}{x}$$

Substitute $h_1 = 66 \text{ m}$ into the first equation:

$$\tan(51^\circ) = \frac{66}{x}$$

Solve for $x$:

$$x = \frac{66}{\tan(51^\circ)}$$ $$x \approx \frac{66}{1.2349}$$ $$x \approx 53.445 \text{ m}$$

Let $h_{add} = h_2 - h_1$. Substitute this into the second equation:

$$\tan(34^\circ) = \frac{h_{add}}{x}$$

Solve for $h_{add}$:

$$h_{add} = x \tan(34^\circ)$$

Substitute the value of $x$ from Step 2:

$$h_{add} \approx 53.445 \text{ m} \times \tan(34^\circ)$$ $$h_{add} \approx 53.445 \text{ m} \times 0.6745$$ $$h_{add} \approx 36.04 \text{ m}$$

The total height of the taller building $h_2$ is the sum of the height of the first building $h_1$ and the additional height $h_{add}$.

$$h_2 = h_1 + h_{add}$$ $$h_2 = 66 \text{ m} + 36.04 \text{ m}$$ $$h_2 = 102.04 \text{ m}$$

Rounding to one decimal place:

$$h_2 \approx 102.0 \text{ m}$$

The height of the taller building is $\text{102.0 m}$.