A ladder 3.5m long is placed against a perpendicular wall with its foot 1.0m from the wall. How far up the wall does the ladder reach?

This problem can be solved using the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Here, the ladder acts as the hypotenuse, the wall is one side of the triangle, and the distance from the wall to the foot of the ladder is the other side.

We know the length of the ladder (hypotenuse, c) is 3.5m and the distance from the wall to the foot of the ladder (one side, a) is 1.0m. We want to find the height up the wall (the other side, b).

The Pythagorean theorem is: a² + b² = c²

We can rearrange this to find b: b² = c² - a²

Substituting the given values: b² = (3.5m)² - (1.0m)² = 12.25m² - 1m² = 11.25m²

Taking the square root of both sides to solve for b (since b² means "b squared"): b = √11.25m² = 3.35m

So, the ladder reaches 3.35m up the wall.