Mandy will need to extend her ladder to 60 feet to reach the roof of her house. To be safe, how far from the base of the house should she place the foot of the ladder?
14 feet
18 feet
20 feet
15 feet
The correct answer and explanation is :
To solve this problem, we need to use the Pythagorean theorem, which applies to right-angled triangles. In this case, the ladder forms the hypotenuse of a right triangle, where the height of the house is one leg, the distance from the base of the house to the foot of the ladder is the other leg, and the ladder itself is the hypotenuse.
Step-by-Step Solution:
We are given:
- The length of the ladder (hypotenuse) = 60 feet
- We need to find the distance from the foot of the ladder to the base of the house (one leg of the right triangle).
- The height of the house (the other leg) is not directly provided in the question, but we can make a reasonable assumption based on standard ladder safety guidelines.
Assumptions:
A typical recommendation for ladder placement is that the distance from the base of the ladder to the wall should be about 1/4 of the ladder’s length for safety reasons. This is a commonly used guideline to ensure the ladder is at a proper angle.
Given the ladder length is 60 feet:
- Distance from the base of the ladder to the wall = 1/4 × 60 = 15 feet.
Thus, the correct answer is 15 feet.
Explanation:
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides (the height of the house and the distance from the foot of the ladder to the wall). The formula is:
[
a^2 + b^2 = c^2
]
Where:
- ( a ) is the height of the house,
- ( b ) is the distance from the base of the ladder to the wall,
- ( c ) is the length of the ladder.
Using the 1/4 guideline ensures the ladder forms a safe angle of about 75 degrees with the ground, minimizing the risk of the ladder slipping or tipping over.
Thus, Mandy should place the foot of the ladder 15 feet away from the base of the house to safely reach the roof.