A 15 m ladder leans against a wall,its bottom is 9m from the wall. How much would the lower end have to be pulled away from the wall so that the top of the ladder would slide down the same amount?
To find out how much the lower end needs to be pulled away from the wall so that the top of the ladder slides down the same amount, we can use similar triangles.
Here's the step-by-step solution:
1. Visualize the situation: Draw a diagram of the ladder leaning against the wall. Label the bottom of the ladder as point A, the top of the ladder as point B, and the point where the bottom end is pulled away from the wall as point C.
2. Identify the given information:
- The distance from the bottom of the ladder to the wall is 9 meters (AC = 9m).
- The length of the ladder is 15 meters (AB = 15m).
3. Identify the unknown:
- The distance the lower end needs to be pulled away from the wall, which is BC.
4. Use similar triangles: In the right triangle formed by the ladder leaning against the wall, the sides are proportional to each other.
According to the similarity of triangles:
BC/AC = AB/BC
5. Substitute the known values:
BC/9 = 15/BC
6. Cross-multiply to solve for BC^2:
BC^2 = 9 * 15
7. Simplify:
BC^2 = 135
8. Take the square root of both sides to find BC:
BC = √135
BC ≈ 11.618
So, the lower end needs to be pulled away from the wall by approximately 11.618 meters so that the top of the ladder slides down the same amount.
To find the distance that the lower end of the ladder needs to be pulled away from the wall, we can use the principle of similar triangles.
Let's consider the situation. We have a right triangle formed by the ladder, the wall, and the ground. The height of the triangle is the height of the wall, which we don't know, and the base is the distance from the wall to the bottom of the ladder, given as 9m. The hypotenuse is the length of the ladder, given as 15m.
Now, if we pull the bottom of the ladder away from the wall by a certain distance, let's call it "x", the new base of the triangle will be 9m + x. The height of the triangle will remain the same, as the top of the ladder is sliding down the same amount as the lower end moves away from the wall.
Using the similarity of the triangles, we can set up the following proportion:
(height of the wall) / (distance from the wall to the bottom of the ladder) = (height of the wall) / (distance from the wall to the new base of the ladder)
Or, in equation form:
(h / 9) = (h / (9 + x))
Now, we can solve the equation for "x" to find how much the lower end of the ladder needs to be pulled away from the wall.
Cross-multiplying the equation, we have:
h(9 + x) = h * 9
Simplifying the equation, we get:
9h + hx = 9h
Subtracting 9h from both sides, we have:
hx = 0
Dividing both sides of the equation by "h", we get:
x = 0
Therefore, to maintain the same slide distance for the top of the ladder, the lower end does not need to be pulled away from the wall.