A ladder is 6m long.How much farther up a wall dose it reach when the foot of the ladder is 2m fro the wall than when it is 3m from the wall?
To find the height the ladder reaches on the wall at two different distances from the wall, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's start by visualizing the problem. We have a ladder leaning against a wall, forming a right-angled triangle. The ladder itself is the hypotenuse, and the distances of the ladder's foot from the wall are the other two sides of the triangle.
Let's call the distance of the ladder's foot from the wall when it is 2m away as 'a', and when it is 3m away as 'b'. The length of the ladder, which is the hypotenuse, is given as 6m.
According to the Pythagorean theorem, we have the equation:
a^2 + x^2 = 6^2
b^2 + y^2 = 6^2
Where 'x' and 'y' represent the height the ladder reaches on the wall at distances 'a' and 'b', respectively.
Since we are interested in finding the difference between the two heights, we can subtract the two equations:
a^2 - b^2 + x^2 - y^2 = 36 - 36
Simplifying further, we have:
(a^2 - b^2) + (x^2 - y^2) = 0
Using the difference of squares formula (a^2 - b^2 = (a + b)(a - b)), we can factor the equation:
(a + b)(a - b) + (x^2 - y^2) = 0
Since we know the values of 'a' and 'b', we can substitute them:
(2 + 3)(2 - 3) + (x^2 - y^2) = 0
-1 + (x^2 - y^2) = 0
(x^2 - y^2) = 1
Now, we need to solve for the difference between the heights, which is represented by (x^2 - y^2). Taking the square root of both sides, we get:
sqrt(x^2 - y^2) = sqrt(1)
x^2 - y^2 = 1
Since we only need the positive value of the difference in heights, we can ignore the negative root.
Therefore, the difference in height that the ladder reaches on the wall when the foot is 2m away compared to when it is 3m away is 1 meter.
Time for ye olde Pythagorean Theorem.
a^2 + b^2 = c^2
2^2 + b^2 = 6^2
4 + b^2 = 36
b^2 = 34
b = 5.83
3^2 + b^2 = 6^2
9 + b^2 = 36
b^2 = 27
b = 5.2