a woman 1.5 tall walks at a rate of 1.2m/s directly away from the street light that is 6m above the street. At what rate is her shadow changing?
To find the rate at which her shadow is changing, we need to use similar triangles and the concept of proportions. Let's consider the situation:
Let 'x' be the distance of the woman from the base of the streetlight, and 'y' be the length of her shadow.
Now, let's set up a proportion between the triangles formed by the woman, her shadow, and the streetlight:
x / y = (x + 1.5) / (y + 6)
Here, we added the height of the woman (1.5 meters) and the height of the streetlight (6 meters) to the corresponding sides of the triangles.
We can cross-multiply to obtain:
x(y + 6) = (x + 1.5)y
Expanding this equation gives us:
xy + 6x = xy + 1.5y
Now, we can solve for 'y', which represents the length of the shadow:
6x = 1.5y
Dividing both sides by 1.5 gives us:
4x = y
So, we found that the length of the shadow 'y' is four times the distance 'x' of the woman from the base of the streetlight.
Now, we need to differentiate this equation with respect to time (t) to find the rate at which her shadow is changing.
Differentiating both sides with respect to 't' gives us:
4(dx/dt) = (dy/dt)
Here, dx/dt represents the rate at which x is changing, and dy/dt represents the rate at which y is changing, which is the rate we are interested in finding.
Since the woman is walking away from the streetlight, dx/dt is given as 1.2 m/s. Thus, we can substitute the known values:
4 * 1.2 = (dy/dt)
Simplifying this gives us:
4.8 = (dy/dt)
Therefore, the rate at which her shadow is changing is 4.8 m/s.
Using similar triangles, if the shadow's length is s and the woman is x away from the light,
s/1.5 = (x+s)/6
That should help, eh?