A building is 20 m away from an observation point. The angle of elevation to the bottom of a window is 27 degrees and to the top is 32 degrees. What is the height of the window?

Let $D$ be the horizontal distance from the observation point to the building, $D = 20$ m.
Let $h_1$ be the height from the ground to the bottom of the window.
Let $h_2$ be the height from the ground to the top of the window.
The angle of elevation to the bottom of the window is $\theta_1 = 27^\circ$.
The angle of elevation to the top of the window is $\theta_2 = 32^\circ$.
The height of the window, $H$, is $h_2 - h_1$.

We use the tangent function, which relates the opposite side (height) to the adjacent side (distance):

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

$$\tan(\theta_1) = \frac{h_1}{D}$$ $$\tan(27^\circ) = \frac{h_1}{20}$$ $$h_1 = 20 \times \tan(27^\circ)$$

Using a calculator, $\tan(27^\circ) \approx 0.509525$.

$$h_1 = 20 \times 0.509525$$ $$h_1 \approx 10.1905 \text{ m}$$

$$\tan(\theta_2) = \frac{h_2}{D}$$ $$\tan(32^\circ) = \frac{h_2}{20}$$ $$h_2 = 20 \times \tan(32^\circ)$$

Using a calculator, $\tan(32^\circ) \approx 0.624869$.

$$h_2 = 20 \times 0.624869$$ $$h_2 \approx 12.4974 \text{ m}$$

The height of the window is the difference between $h_2$ and $h_1$.

$$H = h_2 - h_1$$ $$H = 12.4974 \text{ m} - 10.1905 \text{ m}$$ $$H = 2.3069 \text{ m}$$

Rounding to two decimal places:

$$H \approx 2.31 \text{ m}$$

The height of the window is $\text{2.31 m}$.