To find the height of the building, we can apply trigonometric principles. Specifically, we use the tangent of the angle of elevation. The relationship between the angle of elevation $\theta$, the height of the building $h$, and the length of the shadow $L$ is given by:

$\tan(\theta) = \frac{h}{L}$

Where:

• $\theta = 60^\circ 12'$ (angle of elevation),
• $L = 45.3$ ft (length of the shadow),
• $h$ is the height of the building, which we need to find.

Step 1: Convert the angle to decimal form.

The angle of elevation is $60^\circ 12'$, where $12'$ (minutes) needs to be converted to degrees. Since 1 minute = $\frac{1}{60}$ degree, we have:

$60^\circ 12' = 60^\circ + \frac{12}{60} = 60^\circ + 0.2^\circ = 60.2^\circ$

Step 2: Use the tangent function.

Now we use the tangent function:

$\tan(60.2^\circ) = \frac{h}{45.3}$

Solving for $h$:

$h = 45.3 \times \tan(60.2^\circ)$

Let's compute the height $h$.The height of the building is approximately $79.10$ feet.

Would you like more details or have any further questions?

Here are 5 related questions for deeper understanding:

1. How would the height change if the angle of elevation was 45° instead of 60° 12'?
2. How can we use similar triangles to verify this calculation?
3. How would the length of the shadow change if the building was taller?
4. Can this method be applied to calculate the height of mountains using trigonometry?
5. What is the relationship between the angle of elevation and the length of the shadow?

Tip: Always ensure angles are in the correct units (degrees or radians) when using trigonometric functions in calculations.