A 25-foot ladder is placed against a vertical wall of a building. The base of the ladder is 7 feet from the base of the wall. If the top of the ladder slips 4 feet, how many feet will the base of the ladder slide?
a^2 + b^2 = c^2
a^2 + 7^2 = 25^2
a^2 + 49 = 625
a^2 = 625 - 49
a^2 = 576
a = 24
24 - 4 = 20
20^2 + b^2 = 25^2
400 + b^2 = 625
b^2 = 225
b = 15
15 - 7 = 8
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the ladder forms a right triangle with the wall and the distance it slides forms the other side.
Let's represent the distance the base of the ladder slides as x.
According to the problem, the length of the ladder (hypotenuse) is 25 feet, and the distance between the base of the ladder and the base of the wall is 7 feet.
Using the Pythagorean theorem, we can write the equation:
x^2 + (25 - 4)^2 = 7^2
Simplifying this equation:
x^2 + 21^2 = 7^2
x^2 + 441 = 49
Subtracting 441 from both sides:
x^2 = 49 - 441
x^2 = -392
This equation has no real solutions since the square of any real number cannot be negative.
Therefore, the base of the ladder will not slide since the equation has no real solutions.
To determine how many feet the base of the ladder will slide, let's break down the problem:
1. We are given that the ladder is 25 feet long and it is placed against a vertical wall of a building.
2. The base of the ladder is 7 feet from the base of the wall.
3. The top of the ladder slips 4 feet.
To visualize this situation, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the ladder represents the hypotenuse, the distance the base of the ladder slides represents one side, and the distance the top of the ladder slips represents the other side. Since the ladder is against the wall, it forms a right triangle.
Applying the Pythagorean theorem, we can set up an equation:
(base distance)^2 + (slipped distance)^2 = (ladder length)^2
Substituting the given values:
7^2 + 4^2 = 25^2
49 + 16 = 625
65 = 625
Taking the square root of both sides to solve for the base distance, we find:
(base distance) = √(625 - 16 - 49)
(base distance) = √560
(base distance) ≈ 23.66 feet
Therefore, the base of the ladder will slide approximately 23.66 feet.