We can solve this problem using the Pythagorean theorem, which states that in a right 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.
In this case:
- The length of the ladder is the hypotenuse, which is 41 feet.
- The distance from the base of the ladder to the wall is one leg, which is 9 feet.
- The height (which we want to find) that the ladder reaches up the wall is the other leg.
Let \( h \) be the height up the wall. According to the Pythagorean theorem, we have:
\[ h^2 + 9^2 = 41^2 \]
Calculating \( 9^2 \) and \( 41^2 \):
\[ 9^2 = 81 \] \[ 41^2 = 1681 \]
Now substituting these values into the equation:
\[ h^2 + 81 = 1681 \]
To solve for \( h^2 \), subtract 81 from both sides:
\[ h^2 = 1681 - 81 \] \[ h^2 = 1600 \]
Now take the square root of both sides to find \( h \):
\[ h = \sqrt{1600} \] \[ h = 40 \]
Thus, the top of the ladder reaches 40 feet up the apartment building.