Evan is 1.8 metres tall. He walks between two lampposts that are 5 metres apart and notices his shadow from one of the lampposts just touches the base of the other. He also notices that the ratio of his height to the height of the lamppost is 2 to 5. How far is he from each lamppost?
(I understand so far that the lamppost is 4.5 metres tall and that I have to do something with the triangle shape formed-- but I missed the lesson so please, help?)
To solve this problem, we can use similar triangles and the concept of proportionality.
Let's start by drawing a diagram to visualize the situation:
```
Evan
______________ <--- lamppost B
|
|
| triangle A
|
|
______________ <--- lamppost A
```
We already know that the distance between the lampposts (A and B) is 5 meters and that Evan's height (Evan) is 1.8 meters.
We also know that the ratio of Evan's height to the height of the lamppost is 2/5. This means that the height of lamppost A is (2/5) * 1.8 = 0.72 meters.
Let's call the distance between Evan and lamppost A as x meters. Now we can set up a proportion to find the distance between Evan and lamppost B.
The height of lamppost A forms a similar triangle with Evan's height. Since corresponding sides in similar triangles are proportional, we can write:
(Evan's height) / (Distance between Evan and lamppost A) = (Height of lamppost A) / (Distance between Evan and lamppost B)
So we have:
1.8 / x = 0.72 / (5 - x)
Now, we can solve this equation to find the distance between Evan and each lamppost.
Cross-multiplying the equation, we get:
1.8 * (5 - x) = 0.72 * x
Simplifying, we have:
9 - 1.8x = 0.72x
Combining like terms, we get:
9 = 2.52x
Dividing both sides by 2.52, we find:
x = 9 / 2.52 ≈ 3.57
So, Evan is approximately 3.57 meters away from lamppost A and (5 - 3.57) ≈ 1.43 meters away from lamppost B.