a ladder of length 4m lean against a vertical wall. the foot of the ladder is 2 meter from the wall. a plank that has a length of 5metre rests on a ladder so that one end half way up ladder.

a. how high is the top of a ladder?
b. how high is the top of plank?
c. how far is the motion of the plank from the wall?

Did you draw a diagram?

Did you review the Pythagorean Theorem?
(a) 2^2 + h^2 = 4^2
(b) h/2
(c) I have no idea what the "motion" of the plank is, but ifs top is 1 m from the wall, so if its base is x meters from the wall,
(x-1)^2 + (h/2)^2 = 5^2

To answer these questions, let's break down the information step-by-step:

Step 1: Visualize the scenario
- There is a ladder leaning against a vertical wall.
- The ladder has a length of 4 meters.
- The foot of the ladder is 2 meters from the wall.
- A plank, with a length of 5 meters, rests on the ladder.
- One end of the plank is halfway up the ladder.

Step 2: Find the height of the top of the ladder
- Since the ladder is leaning against the wall, it forms a right-angled triangle with the wall.
- The ladder acts as the hypotenuse, and the distance from the foot of the ladder to the wall acts as the base of the triangle.
- We can use the Pythagorean theorem to find the height of the top of the ladder (opposite side).

a. Using the Pythagorean theorem:
- Let's denote the height of the top of the ladder as h.
- According to the Pythagorean theorem, h^2 = ladder length^2 - base^2.
- Plugging in the values, we get h^2 = 4^2 - 2^2 = 16 - 4 = 12.
- Taking the square root of both sides, we find h = √12 = 2√3 meters.
- Therefore, the height of the top of the ladder is 2√3 meters.

Step 3: Find the height of the top of the plank
- As mentioned earlier, one end of the plank is halfway up the ladder.
- This means the height of the top of the plank is half the height of the ladder's top.

b. Therefore, the height of the top of the plank is (1/2) * 2√3 = √3 meters.

Step 4: Find the distance of the plank from the wall
- This can be calculated by subtracting the distance of the foot of the ladder from the length of the plank.

c. Therefore, the distance of the plank from the wall is 5 - 2 = 3 meters.

Summary of results:
a. The height of the top of the ladder is 2√3 meters.
b. The height of the top of the plank is √3 meters.
c. The distance of the plank from the wall is 3 meters.

To solve this problem, we can use the concept of similar triangles. Let's break down the problem step by step:

a. How high is the top of the ladder?
To find the height of the top of the ladder, we can establish a ratio between similar triangles. Let's define the height of the top of the ladder as h.

Using the information given, we have a right-angled triangle formed by the ladder, the wall, and the ground. The base of this triangle is 2m, and the hypotenuse is 4m (length of the ladder). By using Pythagoras' theorem, we can find the height of the ladder (h).

Using Pythagoras' theorem:
a² + b² = c²
(2²) + (h²) = (4²)

Simplifying the equation:
4 + h² = 16
h² = 16 - 4
h² = 12
h = √12
h ≈ 3.46m

Therefore, the top of the ladder is approximately 3.46m high.

b. How high is the top of the plank?
Since one end of the plank is placed halfway up the ladder, we can consider another right-angled triangle formed by the ladder, the plank, and the ground. Let's define the height of the top of the plank as p.

Now, we can establish a ratio between the height of the ladder and the height of the plank, as they are similar triangles:

h / p = (length of the ladder) / (length of the plank)
3.46 / p = 4 / 5

Cross-multiplying the equation:
5 * 3.46 = 4 * p
p ≈ 4.36m

Therefore, the top of the plank is approximately 4.36m high.

c. How far is the bottom of the plank from the wall?
Since we know that one end of the plank is placed halfway up the ladder, we can consider another right-angled triangle formed by the plank, the wall, and the ground. Let's define the distance of the bottom of the plank from the wall as d.

Using the similar triangles concept once again:

h / d = (length of the ladder) / (length of the plank)
3.46 / d = 4 / 5

Cross-multiplying the equation:
5 * 3.46 = 4 * d
d ≈ 4.36m

Therefore, the distance of the bottom of the plank from the wall is approximately 4.36m.