To solve this problem, we need to use the Pythagorean theorem, since the building, its shadow, and the line from the top of the building to the tip of the shadow form a right triangle.

The Pythagorean theorem states: $a^2 + b^2 = c^2$ where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse.

• The length of the shadow is $30 \, \text{m}$, which is one leg of the triangle.
• The distance from the top of the building to the tip of the shadow is $37 \, \text{m}$, which is the hypotenuse.
• We need to find the height of the building, which is the other leg.

Let $h$ be the height of the building. Using the Pythagorean theorem: $h^2 + 30^2 = 37^2$ Now, let's solve for $h$.

$h^2 + 900 = 1369$ $h^2 = 1369 - 900$ $h^2 = 469$ $h = \sqrt{469}$ $h \approx 21.7 \, \text{m}$

The height of the building is approximately 21.7 meters.

Would you like further details or explanations? Here are five related questions you could explore:

1. How do you determine if a triangle is a right triangle using side lengths?
2. What are some real-life applications of the Pythagorean theorem?
3. How would the problem change if the shadow length was longer than the distance to the tip?
4. Can the Pythagorean theorem be used for non-right triangles? If not, what other methods are there?
5. How would you calculate the height of the building if you were only given angles?

Tip: When using the Pythagorean theorem, always ensure the hypotenuse is correctly identified as the longest side of the triangle.