A street light is at the top of a 18 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 6 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 find the rate at which the tip of the woman's shadow is moving, we can use similar triangles and the concept of related rates.

Let's define the following variables:
- Let x be the distance between the woman and the base of the pole.
- Let y be the length of the woman's shadow cast by the street light.
- Let z be the distance between the tip of the woman's shadow and the base of the pole.

We want to find the rate at which z is changing when x = 45 ft.

Since the street light is at the top of an 18 ft pole, the length of the pole is the sum of the woman's height and the length of her shadow:
18 ft = 6 ft + y

Simplifying the equation, we find:
y = 12 ft

Using similar triangles, we know that the ratio of the lengths of corresponding sides in similar triangles is constant. This gives us the following proportion:
(y + z) / x = y / z

Substituting known values, we have:
(12 + z) / 45 = 12 / z

Cross-multiplying, we get:
12z = (12 + z) * 45

Expanding and rearranging the equation, we have:
12z = 540 + 45z

Simplifying further:
33z = 540

Dividing both sides by 33, we find:
z = 16.36 ft (approximately)

Now, to find the rate at which z is changing, we take the derivative of both sides of the equation with respect to time:
d/dt(z) = d/dt(16.36)

The derivative of z with respect to time represents the rate at which z is changing. Since the woman is walking away from the pole at a speed of 6 ft/sec, dx/dt (the rate at which x is changing) is 6 ft/sec.

Taking the derivative of z = 16.36 with respect to time, we get:
dz/dt = 0

Therefore, the rate at which the tip of the woman's shadow is moving when she is 45 ft from the base of the pole is 0 ft/sec.

Draw a triangle.

If the woman is x feet from the pole, and her shadow extends an additional y feet,

18/(x+y) = 6/y
18y = 6(x+y)
18x = 6x + 6y
12x = 6y
2x = y

2 dx/dt = dy/dt
2(6) = dy/dt

Now, the distance of the shadow tip from the pole is x+y, so its speed is dx/dt + dy/dt = 6+12 = 18 ft/sec, regardless of the distance.