This problem involves a right triangle, where:

• The rake's length (55 inches) represents the hypotenuse.
• The height at which the rake touches the wall (44 inches) is one leg.
• We are tasked with finding the distance from the base of the wall to the bottom of the rake, which is the other leg.

Using the Pythagorean theorem:

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

Where:

• $a$ is the unknown distance (the base of the triangle).
• $b = 44$ inches (the height of the triangle).
• $c = 55$ inches (the hypotenuse).

We can rearrange the equation to solve for $a$:

$a^2 = c^2 - b^2$ $a^2 = 55^2 - 44^2$ $a^2 = 3025 - 1936$ $a^2 = 1089$ $a = \sqrt{1089} = 33$

So, the bottom of the rake is 33 inches from the base of the wall.

Would you like a more detailed explanation of how the Pythagorean theorem works? Here are some related questions:

1. What are the conditions under which the Pythagorean theorem applies?
2. Can the Pythagorean theorem be used to calculate distances in three-dimensional space?
3. How do we calculate the length of the hypotenuse if the legs of the triangle are given?
4. How do you apply the converse of the Pythagorean theorem?
5. What is a real-life example where you would need to apply the Pythagorean theorem?

Tip: Always double-check your values when using the Pythagorean theorem to avoid small calculation errors.