Question

a 10ft ladder leans against a wall, the bottom of the ladder is 6ft from the wall,how much would the lower end have to be pulled away so that the top end would be pulled down the same amount?

Answer 1

Right now the triangle formed by the ladder is a 3-4-5 right triangle, with an hypotenuse of 10 and one leg of 6 -- therefore the other leg (the height up the wall) must be 8. Adding x feet to the leg along the ground, and subtracting the same x feet from the height along the wall.

GUESS: Adding 2 feet to the length along the ground will change this into a different 3-4-5 right triangle, with the bottom leg adding 2 feet and the height along the wall subtracting 2 feet.

So the answer is: The lower end would have to be pulled away from the wall by 2 feet so that the height up the wall would be reduced by 2 feet.

Here is the math:

6^2+ 8^2= 10^2

1) Add x feet to the length along the ground and subtract x feet from the height up the wall.
( 6 + x )^2+ ( 8 - x )^2= 100

2) Expand the polynomials
36 + 12x + x^2 + 64-16x + x^2= 100

3) Combine like terms
2x^2- 4x + 100 = 100

4) Subtract 100 from both sides
2x^2-4x = 0

5) Divide each side by 2
x^2- 2x = 0

6) Add 2x to each side
X^2= 2x

7) Divide both sides by x
x = 2

Answer: If you pull the ladder out 2 more feet from the wall, the top of the ladder will drop by 2 feet