A ladder is leaning against a building so that the distance from the ground to the top of the ladder is one foot less than the length of the ladder. Find the length of the ladder if the distance from the bottom of the ladder to the building is 5 feet.
AAAaannndd the bot gets it wrong yet again!
f the length of the ladder is x, we have
5^2 + (x-1)^2 = x^2
x = 13
You could save some calculation if you just remember the 5-12-13 Pythagorean triple.
Let's assume that the length of the ladder is "x" feet.
According to the problem, the distance from the ground to the top of the ladder is one foot less than the length of the ladder. So, the height of the ladder will be (x - 1) feet.
Also, it is given that the distance from the bottom of the ladder to the building is 5 feet.
We can create a right-angled triangle using the ladder, the building, and the ground. The ladder represents the hypotenuse, the height represents one of the legs, and the distance from the bottom of the ladder to the building represents the other leg.
Using the Pythagorean theorem, we can write:
(x - 1)^2 = 5^2 + 5^2
Expanding the equation:
x^2 - 2x + 1 = 25 + 25
x^2 - 2x + 1 = 50
Rearranging the equation:
x^2 - 2x - 49 = 0
We can solve this quadratic equation using factoring or the quadratic formula. Let's use the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -2, and c = -49.
Plugging these values into the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4(1)(-49))) / (2(1))
Simplifying:
x = (2 ± √(4 + 196)) / 2
x = (2 ± √200) / 2
x = (2 ± 10√2) / 2
Simplifying further:
x = 1 ± 5√2
We have two possible solutions: x = 1 + 5√2 and x = 1 - 5√2.
Since the length of the ladder cannot be negative, we will consider the positive solution.
Therefore, the length of the ladder is 1 + 5√2 feet.
To find the length of the ladder, we can use the Pythagorean theorem.
Let's denote the length of the ladder as 'L'. According to the problem, the distance from the bottom of the ladder to the building is 5 feet and the distance from the ground to the top of the ladder is 1 foot less than the length of the ladder.
Using the Pythagorean theorem, we have:
(Length of ladder)^2 = (Distance from the bottom of the ladder to the building)^2 + (Distance from the ground to the top of the ladder)^2
Substituting the given values:
L^2 = 5^2 + (L-1)^2
Expanding the equation:
L^2 = 25 + L^2 - 2L + 1
Combining like terms:
0 = 26 - 2L
Rearranging the equation to solve for L:
2L = 26
L = 26/2
L = 13
Therefore, the length of the ladder is 13 feet.