To solve this problem, we can use the concept of similar triangles. Let's break down the solution step by step:

1. Let's consider the vertical pole and its shadow. We can create a right triangle with the pole as the height and the shadow as the base. Using the Pythagorean theorem, we can find the length of the hypotenuse (the distance from the top of the pole to the end of the shadow):

The length of the hypotenuse = √(height^2 + base^2)
= √(6^2 + 4^2)
= √(36 + 16)
= √52
= 2√13

2. Now, let's consider the tower and its shadow. We can create another right triangle with the height of the tower as the height and the shadow of the tower as the base. Using the same logic as before, we can find the length of the hypotenuse:

The length of the hypotenuse = √(height^2 + base^2)
= √(height^2 + 25^2)
= √(height^2 + 625)

3. Since the two triangles are similar, their corresponding sides are proportional. This means that the ratio of the height of the pole to the height of the tower is equal to the ratio of the length of the hypotenuse of the pole triangle to the length of the hypotenuse of the tower triangle:

height of pole / height of tower = length of hypotenuse of pole / length of hypotenuse of tower

6 / height of tower = 2√13 / √(height^2 + 625)

4. Cross-multiplying and simplifying the equation, we get:

6 * √(height^2 + 625) = 2√13 * height of tower

5. Squaring both sides of the equation to eliminate the square roots, we get:

36 * (height^2 + 625) = 4 * 13 * (height^2)

36 * height^2 + 22500 = 52 * height^2

16 * height^2 = 22500

height^2 = 22500 / 16

height^2 = 1406.25

height = √1406.25

height ≈ 37.5 m

Therefore, the height of the tower is approximately 37.5 m, which corresponds to option (c).