To find the height of the tree using the Pythagorean theorem, we can set up a right triangle with the tree's height as one leg, the distance to the end of the shadow as the other leg, and the shadow's length as the hypotenuse.
Here's how we can solve it step by step:
1. Let the height of the tree be \( h \), the distance to the end of the shadow be 15 meters, and the shadow's length be 3.5 meters.
2. Apply the Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs of the triangle and \( c \) is the hypotenuse.
3. Substitute the values into the formula:
\[ h^2 + 15^2 = 3.5^2 \]
4. Solve for the height of the tree:
\[ h^2 + 225 = 12.25 \]
\[ h^2 = 12.25 - 225 \]
\[ h^2 = 191.75 \]
5. Take the square root of both sides to find the height:
\[ h = \sqrt{191.75} \]
\[ h \approx 13.84 \text{ meters} \]
Therefore, the height of the tree is approximately 13.84 meters.
