A street light is at the top of a 14 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 ft from the base of the pole?
To determine how fast the tip of the woman's shadow is moving when she is 45 ft from the base of the pole, we can use related rates. We need to establish a relationship between the distances involved in the problem.
Let's consider the similar triangles formed by the pole, the woman, and her shadow. We can set up the proportion:
14 ft (pole height) / x ft (length of the woman's shadow) = (14 ft + 6 ft) / (x + y) ft
Where y ft represents the distance of the woman from the base of the pole and x ft is the length of her shadow at this moment.
Simplifying the proportion:
14 / x = 20 / (x + y)
Next, differentiate both sides of the equation with respect to time (t):
d/dt[14/x] = d/dt[20/(x + y)]
To find d(x)/dt, the rate at which the shadow is changing, we need to determine dx/dt. To find dy/dt, the rate at which the woman moves away from the pole, we need to establish a relationship between y, x, and the distance of the woman from the pole. This relationship can be expressed as:
y^2 + x^2 = d^2
Where d is the distance of the woman from the pole (45 ft in this case).
Differentiating with respect to time once again:
2y(dy/dt) + 2x(dx/dt) = 0
We already know dy/dt (the woman's speed), which is 7 ft/sec, and we need to find dx/dt when x = 45 ft.
Rearrange this equation to solve for dx/dt:
2y(dy/dt) = -2x(dx/dt)
dx/dt = -(2y(dy/dt)) / (2x)
Substituting the given values:
dx/dt = -(2 * 45 ft * 7 ft/sec) / (2 * 45 ft)
Simplifying further:
dx/dt = -7 ft/sec
Therefore, the tip of the woman's shadow is moving at a rate of 7 ft/sec in the opposite direction as the woman when she is 45 ft from the base of the pole.
Using similar triangles, the distance s of the shadow's tip when she is x ft from the pole, is
(s-x)/6 = s/14
s = 7/4 x
ds/dt = 7/4 dx/dt
Check my math, and then plug in your numbers.
Note that the speed does not depend on her distance x