This problem involves finding the distance between the tops of two poles of different heights standing vertically on level ground. We are given the heights of the poles and the distance between their bases. This scenario can be modelled using geometry, specifically a right-angled triangle.

Poles Setup and Visualization

Imagine two vertical poles, Pole A and Pole B, standing on flat ground.

• Pole A height: 12 m
• Pole B height: 20 m
• Distance between the bases of Pole A and Pole B: 15 m

We need to find the direct distance between the top of Pole A and the top of Pole B.

To solve this, we can form a right-angled triangle. Draw a horizontal line from the top of the shorter pole (Pole A) parallel to the ground until it meets the taller pole (Pole B). Let's call the point where this line meets Pole B as point C.

This creates a rectangle with the base distance and the height of the shorter pole, and a right-angled triangle above this rectangle. The vertices of the right-angled triangle are:

• The top of Pole A.
• The top of Pole B.
• Point C on Pole B (at the same height as the top of Pole A).

Calculating Distance Between Tops

Now, let's identify the sides of this right-angled triangle:

• Horizontal Leg: The distance between the bases of the poles is 15 m. This is the same as the horizontal distance between the tops of the poles. So, one leg of the triangle is 15 m.
• Vertical Leg: This is the difference in height between the two poles. The height difference is $20 \text{ m} - 12 \text{ m} = 8 \text{ m}$. This is the other leg of the triangle.
• Hypotenuse: The distance between the tops of the poles is the hypotenuse of the triangle.

Applying the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$). The formula is:

$$ a^2 + b^2 = c^2 $$

In our case:

• $a = 8$ m (the vertical leg, height difference)
• $b = 15$ m (the horizontal leg, distance between bases)
• $c$ is the distance between the tops.

Substitute the values into the formula:

$$ (8 \text{ m})^2 + (15 \text{ m})^2 = c^2 $$

Calculate the squares:

$$ 64 \text{ m}^2 + 225 \text{ m}^2 = c^2 $$

Add the results:

$$ 289 \text{ m}^2 = c^2 $$

To find $c$, take the square root of both sides:

$$ c = \sqrt{289 \text{ m}^2} $$

$$ c = 17 \text{ m} $$

Therefore, the distance between the tops of the two poles is 17 meters.