A lamppost is 20 feet tall and a man is standing 30 feet away casting a 10 foot shadow. How tall is the man?
draw a diagram ... it's a proportion ... similar triangles
short for a man ... maybe a boy
5 feet right?
To determine the height of the man, we can use similar triangles and the concept of proportions. Here's how to solve it step by step:
1. Start by drawing a diagram to visualize the problem. Draw a vertical line to represent the lamppost and another line to represent the man. Include the shadow cast by the man.
```
X
-------------
| |
| |
| |
| | <- Lamppost (20 ft)
| |
| |
|M |
| |
| |
--------------
```
2. Identify the similar triangles. In this case, we have two similar triangles: one formed by the lamppost and its shadow, and the other formed by the man and his shadow. Let's label the height of the lamppost as L and its shadow as S. Similarly, label the height of the man as M and his shadow as X.
```
X
-------------
| |
| |
| |
| |<- Lamppost (L = 20 ft)
| |<- Lamppost Shadow (S = 10 ft)
|M |
| |
| |
--------------
```
3. Set up the proportionality equation. Since the triangles are similar, the ratio of corresponding sides will be equal. We can set up the following proportion:
```
L / S = M / X
```
4. Substitute the given values into the equation. We know that the height of the lamppost (L) is 20 ft and its shadow (S) is 10 ft. We need to find the height of the man (M).
```
20 / 10 = M / X
```
5. Solve for M. Cross-multiply to get:
```
20X = 10M
```
Divide both sides by 10 to isolate M:
```
2X = M
```
6. Substitute the given value for the man's shadow (X), which is 30 ft:
```
2 * 30 = M
```
7. Calculate:
```
M = 60
```
The man's height is 60 feet.