a 6 foot person is standing x feet away form a 10 foot lamppost. What is the distance d from the base of the lamppost to the end of the persons shadow, espressed as a function of x.
To find the distance (d) from the base of the lamppost to the end of the person's shadow, expressed as a function of x, we can use similar triangles.
Let's call the height of the lamppost h and the length of the person's shadow s.
Using the concept of similar triangles, we can set up a proportion:
(person's height) / (person's shadow length) = (lamppost height) / (distance from the base of the lamppost to the end of the person's shadow)
We know the person's height is 6 feet, the lamppost height is 10 feet, and the distance from the person to the lamppost is x feet.
So, the proportion can be written as:
6 feet / s = 10 feet / (s + x)
Now, we can solve for s by cross-multiplying and simplifying the equation:
6(s + x) = 10s
6s + 6x = 10s
6x = 10s - 6s
6x = 4s
Now, divide both sides of the equation by 4 to isolate s:
s = (6x) / 4
s = 1.5x
Therefore, the distance from the base of the lamppost to the end of the person's shadow, expressed as a function of x, is d = s + x:
d = 1.5x + x
d = 2.5x
To find the distance from the base of the lamppost to the end of the person's shadow, we can use similar triangles. Let's denote the distance from the base of the lamppost to the end of the person's shadow as "d."
We have a right triangle formed by the person, the end of their shadow, and the top of the lamppost. The height of this triangle is the person's height, which is 6 feet. The length of the person's shadow is x feet, and the height of the lamppost is 10 feet.
Since we have similar triangles, we can set up a proportion:
(person's height) / (length of person's shadow) = (height of lamppost) / (distance from base of lamppost to end of shadow)
Using the given measurements, we can write:
6 / x = 10 / d
Cross multiplying, we have:
6d = 10x
Now, we can solve for d:
d = (10x) / 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:
d(x) = (10x) / 6