A Ladder AB, of length 13m, rests against a vertical wall with its foot on a horizontal floor at a distance of 5meters from the wall. When the top of the ladder slips down a distance x meters on the wall, the foot of the ladder moves out x meters. Find x.
(I couldn't find this answer, can anyone please please help me. With the math equation too? Not the answer only please. So I know what to do it. Thx.)
To solve this problem, we can use the Pythagorean theorem. According to the theorem, in a right-angled 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 AB forms a right-angled triangle with the vertical wall and the horizontal floor. Let's label the distance on the wall that the ladder slips down as x.
We know that the length of the ladder is 13 meters, and the foot of the ladder moves out x meters. Therefore, the length of the ladder touching the wall is (13 - x) meters.
Using the Pythagorean theorem, we can write the equation:
(x^2) + ((13 - x)^2) = 5^2
Simplifying the equation:
x^2 + (169 - 26x + x^2) = 25
Combining like terms:
2x^2 - 26x + 144 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 2, b = -26, and c = 144.
Plugging in these values in the formula, we get:
x = (-(-26) ± √((-26)^2 - 4 * 2 * 144)) / (2 * 2)
Simplifying further:
x = (26 ± √(676 - 1152)) / 4
x = (26 ± √(-476)) / 4
Since the square root of a negative number is not a real number, it means that no real value of x satisfies the equation. Therefore, there is no solution to this problem.
To find the value of x, we can use the Pythagorean theorem and set up an equation based on the given information.
Let's assume x is the distance the top of the ladder slips down on the wall. Since the ladder is initially resting against the wall, the vertical distance covered by the top of the ladder down the wall will be x.
Given:
Length of the ladder, AB = 13 meters
Distance of the foot of the ladder from the wall = 5 meters
Now, let's set up the equation using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
Using this theorem, we can write the equation as follows:
x^2 + (5 + x)^2 = 13^2
Let's solve this equation to find the value of x.
Expanding the equation, we get:
x^2 + (25 + 10x + x^2) = 169
Combining like terms, we have:
2x^2 + 10x + 25 = 169
Rearranging the equation:
2x^2 + 10x + 25 - 169 = 0
2x^2 + 10x - 144 = 0
Now, we can solve this quadratic equation either by factoring or using the quadratic formula:
Using factoring, we can observe that 2x^2 - 12x - 12x - 144 = 0.
Factoring out common terms, we get:
2x(x - 6) - 12(x - 6) = 0
(x - 6)(2x - 12) = 0
Setting each factor equal to zero:
x - 6 = 0 or 2x - 12 = 0
x = 6 or x = 6
Therefore, the value of x is 6 meters.
Actually, it is calculus of differentials.
C^2=a^2 + b^2
Take the differential
2C dC=2a da + 2b db
but dC is zero (C is held constant), so
da=- (b/a)db
so whatever one side slips down the wall, so the other side slips along the floor in the ratio cited.
It is like triangle.
a^2+b^2=c^2
you know that one of the side and hyp. so, you can find the ohter by pluging in like this:
5^2+b^2= 13^2
25+b^2=169
b^2=144
b= 12