a 6' person is standing x feet away form a 10' lamppost. what is the distance d from the base of the lamppost to the end of the person's shadow, expressed as a function of x.
To solve this problem, we can use similar triangles. Let's first draw a diagram to visualize the situation:
```
C
|\
| \
| \
| \
| \
| \
| \
A-------B
```
Here, A represents the base of the lamppost, B represents the end of the person's shadow, C represents the top of the person's head, and the line connecting A and C represents the person's height of 6'. The line connecting B and C represents the shadow.
We know that AB represents the person's height, so AB = 6'. The distance AC represents the height of the lamppost, which is given as 10'.
Now, let's consider the similar triangles ABD and ABC. These triangles are similar because they share angle A and angle B. Therefore, the corresponding sides are proportional.
The length of AD (distance from the base of the lamppost to the end of the person's shadow, which is the distance we want to find) corresponds to the length of AC (height of the lamppost, which is 10'). Similarly, the length of BD (the length of the person's shadow) corresponds to the length of BC (the distance from the person to the lamppost).
Now, we can set up a proportion to find the value of AD in terms of x:
AD/AC = BD/BC
Substituting the known values:
AD/10' = x/6'
Cross-multiplying:
AD * 6' = 10' * x
Simplifying:
AD = (10' * x) / 6'
Therefore, the distance from the base of the lamppost to the end of the person's shadow, expressed as a function of x, is AD = (10' * x) / 6'.