A ladder 10 ft long leans against a wall. The bottom of the ladder is 6 ft from the wall. How much would the lower end of the ladder have to be pulled away so that the top end would be pulled down by 3 ft?
draw the picture
h is height along the wall
x is the distance along the floor from the wall
x^2+h^2=10^2
36+h^2=100
h= 8
2xdx+2hdh=0
dx=-h/x dh
dh=-3, x=6, h=8, dx= 8*3/6= 4ft
check that.
To find out how much the lower end of the ladder needs to be pulled away, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the ladder forms the hypotenuse of a right triangle, and the distance from the wall to the lower end of the ladder forms one of the sides.
Let's denote the distance the lower end needs to be pulled away as x. Thus, the distance from the wall to the lower end of the ladder would then be 6 + x.
Using the Pythagorean theorem, we can set up the equation:
(6 + x)^2 = 10^2 - 3^2
Simplifying this equation, we have:
36 + 12x + x^2 = 100 - 9
Rearranging the equation, we get:
x^2 + 12x - 73 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Factoring might not be straightforward in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing the equation to the quadratic form ax^2 + bx + c = 0, we have a = 1, b = 12, and c = -73. Plugging these values into the quadratic formula, we get:
x = (-12 ± √(12^2 - 4(1)(-73))) / (2(1))
x = (-12 ± √(144 + 292)) / 2
x = (-12 ± √436) / 2
x ≈ (-12 ± 20.87) / 2
Using the plus-minus symbol, we have two possible values for x:
x ≈ (-12 + 20.87) / 2 ≈ 4.44
x ≈ (-12 - 20.87) / 2 ≈ -16.44
Since the distance cannot be negative, we can ignore the negative value. Therefore, the lower end of the ladder needs to be pulled away by approximately 4.44 ft in order for the top end to be pulled down by 3 ft.