a painter leans a 25 foot ladder against a building. The distance from the ground to the top of the ladder is 17 feet more than the distance from the building to the base of the ladder. Find the distance from the building to the base of the ladder and the distance from the ground to the top of the ladder
To solve this problem, let's denote the distance from the building to the base of the ladder as "x". Therefore, the distance from the ground to the top of the ladder will be "x + 17" since it is 17 feet more than the distance from the building to the base.
According to the problem, we know that the ladder is 25 feet long. We can picture the ladder, building, and ground forming a right triangle, with the ladder being the hypotenuse.
Using the Pythagorean theorem, we can write the equation:
x^2 + (x + 17)^2 = 25^2
Simplifying this equation will allow us to find the values of x and x + 17.
Expanding and combining like terms:
x^2 + (x^2 + 34x + 289) = 625
Combining like terms:
2x^2 + 34x + 289 = 625
Rearranging to bring all terms to one side:
2x^2 + 34x + 289 - 625 = 0
Simplifying further:
2x^2 + 34x - 336 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Since it may not be easy to factor this equation, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = 34, and c = -336:
x = (-(34) ± √((34)^2 - 4(2)(-336))) / (2(2))
Simplifying:
x = (-34 ± √(1156 + 2688)) / 4
x = (-34 ± √3844) / 4
x = (-34 ± 62) / 4
So the two possible values for x are:
x₁ = (-34 + 62) / 4 = 7
x₂ = (-34 - 62) / 4 = -24
Since we are dealing with distances, the negative value for x, x₂ = -24, does not make sense in this context.
Therefore, the distance from the building to the base of the ladder is x₁ = 7 feet.
The distance from the ground to the top of the ladder is x₁ + 17 = 7 + 17 = 24 feet.
Let us take AC=Length of a ladder
BC = Distance between ladder base and wall of a building.
AB = Distance which we have to find out.( A point where ladder rests)
Triangle formed is angle ABC which is right angle triangle.
In angle ABC , AB^2 + BC^2 = AC^2
AB^2 = AC^2 - BC^2
AB^2 = 25^2 - 7^2 = 576
AB = sqrt [ 576 ]
AB = 24 ft high ladder reach high on the building.