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
Make a sketch and notice the similar triangles.
by ratio:
d/10 = (d-x)/6
6d = 10d - 10x
-4d = -10x
d = (5/2)x
To solve this problem, we can use similar triangles.
Let the height of the person be P, the height of the lamppost be L, and the distance from the base of the lamppost to the end of the person's shadow be d.
Since the triangles formed by the person, the lamppost, and their shadows are similar, we have:
(P + d) / P = L / x
Cross-multiplying, we get:
x(P + d) = LP
Expanding, we have:
xP + xd = LP
Rearranging the equation:
d = (LP - xP) / x
Substituting the given values L = 10 feet and P = 6 feet, the equation becomes:
d = (10(6) - x(6)) / x
Simplifying further:
d = (60 - 6x) / x
So, the distance d from the base of the lamppost to the end of the person's shadow is expressed as the function:
d(x) = (60 - 6x) / x
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 assume that the person's height of 6 feet represents the length of the shadow, and d represents the distance from the base of the lamppost to the end of the shadow.
We have a right triangle formed by the person, the lamppost, and the ground. The height of this triangle is the person's height (6 feet), and the hypotenuse is the length of the shadow (also 6 feet). The base of the triangle is the distance from the base of the lamppost to the end of the shadow (d feet).
Using the concept of similar triangles, we can set up the following proportion:
(person's height) / (length of shadow) = (distance from base of lamppost to end of the shadow) / (length of shadow)
6 / 6 = d / 10
Simplifying the equation:
1 = d / 10
We can cross-multiply and solve for d:
d = 10
Therefore, the distance d from the base of the lamppost to the end of the person's shadow, expressed as a function of x (the distance from the person to the lamppost), is simply 10 feet.