Answer:
4 ft higher
Step-by-step explanation:
Since the ladder is 10 ft long and its top is 6 feet high(above the ground), we find the distance of its base from the wall since these three (the ladder, wall and ground) form a right angled triangle. Let d be the distance from the wall to the ladder.
So, by Pythagoras' theorem,
10² = 6² + d² (the length of the ladder is the hypotenuse side)
d² = 10² - 6²
d² = 100 - 36
d² = 64
d = √64
d = 8 ft
Since the ladder is moved so that the base of the ladder travels toward the wall twice the distance that the top of the ladder moves up.
Now, let x be the distance the top of the ladder is moved, the new height of top of the ladder is 6 + x. Since the base moves twice the distance the top of the ladder moves up, the new distance for our base is 8 - 2x(It reduces since it gets closer to the wall).
Now, applying Pythagoras' theorem to the ladder with these new lengths, we have
10² = (6 + x)² + (8 - 2x)²
Expanding the brackets, we have
100 = 36 + 12x + x² + 64 - 32x + 4x²
collecting like terms, we have
100 = 4x² + x² + 12x - 32x + 64 + 36
100 = 5x² - 20x + 100
Subtracting 100 from both sides, we have
100 - 100 = 5x² - 20x + 100 - 100
5x² - 20x = 0
Factorizing, we have
5x(x - 4) = 0
5x = 0 or x - 4 = 0
x = 0 or x = 4
The top of the ladder is thus 4 ft higher
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