Verified Answer :
- Assign the ladder length, base distance, and window height to variables.
- Utilize the Pythagorean theorem, \(h^2 + b^2 = L^2\), to solve for the maximum height \(h\) the ladder can reach.
- Calculate \(h = \sqrt{25^2 - 15^2} = 20 \text{ ft}\).
- Compare the result to the target window height to conclude that the ladder does not reach the window, since \(\boxed{20 \text{ ft} < 21.5 \text{ ft}}\).
Explanation
- Define variables and given conditions.
Let \(L\) be the length of the ladder, \(b\) be the distance from the base of the ladder to the wall, and \(h\) be the vertical height reached by the ladder up the wall. The wall is assumed to be perpendicular to the ground, forming a right-angled triangle where the ladder acts as the hypotenuse. We are given:
- The length of the ladder, \(L = 25 \text{ ft}\).
- The distance of the base from the wall, \(b = 15 \text{ ft}\).
- The target height of the window, \(h_{window} = 21.5 \text{ ft}\).
We need to determine if the ladder's reach \(h\) is sufficient to reach the window, i.e., whether \(h \ge 21.5 \text{ ft}\).
- Apply the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle with sides \(a\), \(b\) and hypotenuse \(c\), the relationship is \(a^2 + b^2 = c^2\). In our case:
- The height \(h\) is one leg (\(a = h\)).
- The distance from the base to the wall is the other leg (\(b = 15 \text{ ft}\)).
- The ladder itself is the hypotenuse (\(c = L = 25 \text{ ft}\)).
Substituting the known values into the theorem:
\[h^2 + 15^2 = 25^2\]
\[h^2 + 225 = 625\]
\[h^2 = 625 - 225\]
\[h^2 = 400\]
\[h = \sqrt{400} = 20 \text{ ft}\]
The ladder reaches a vertical height of \(20 \text{ ft}\) up the wall.
- Compare the calculated height with the window height.
The height reached by the ladder is \(h = 20 \text{ ft}\). The window is located at a height of \(21.5 \text{ ft}\). Since \(20 \text{ ft} < 21.5 \text{ ft}\), the ladder does not reach the window. The ladder is too short by \(1.5 \text{ ft}\). Therefore, the answer is no, it does not reach.
Examples
This geometry application is useful for everyday tasks, such as determining if a ladder is safe or appropriate for maintenance work around a house. By understanding the geometric constraints, you can ensure the ladder's base is positioned correctly to achieve the required height safely. It serves as a practical lesson in applying the Pythagorean theorem to real-world scenarios, preventing accidents and ensuring equipment is used effectively for its intended height requirements.