This problem can be solved using the Pythagorean theorem, which applies to right-angled triangles. The ladder forms a right triangle where:

• The ladder's length (12 ft) is the hypotenuse of the right triangle.
• The distance from the bottom of the ladder to the wall (3 ft) is one of the legs of the triangle.
• The height at which the ladder touches the wall is the other leg of the triangle.

We can use the Pythagorean theorem:

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

where:

• $a$ is the distance from the bottom of the ladder to the wall (3 ft),
• $b$ is the height the ladder reaches on the wall (what we're solving for),
• $c$ is the length of the ladder (12 ft).

Plugging in the known values:

$3^2 + b^2 = 12^2$

$9 + b^2 = 144$

Now, subtract 9 from both sides:

$b^2 = 144 - 9$

$b^2 = 135$

Taking the square root of both sides:

$b = \sqrt{135} \approx 11.62 \, \text{ft}$

So, the ladder touches the wall at approximately 11.62 feet above the ground.

Let me know if you need any further details!

Related Questions:

1. How would the result change if the ladder's length were increased to 15 feet, keeping the bottom distance at 3 feet?
2. What happens if the ladder is placed further away from the wall? How would that affect the height at which it touches the wall?
3. How can this method be applied to other real-world scenarios, such as setting up a ramp?
4. What is the importance of the Pythagorean theorem in architecture and construction?
5. How would you calculate the angle the ladder makes with the ground using trigonometry?

Tip:

When using the Pythagorean theorem, always ensure that you're dealing with a right triangle, where the two legs are perpendicular to each other.