A street light is at the top of a 13 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 feet from the base of the pole?

as always, draw a diagram. Using similar triangles, you can see that if her shadow has length s, then when she is x feet from the pole,

s/6 = (x+s)/13
6(x+s) = 13s
6x+6s = 13s
s = 6/7 x
so at any time,
ds/dt = 6/7 dx/dt
so the shadow's length is increasing at 6/7 * 8 = 48/7 ft/s
now add that to the speed of the woman, and the tip of the shadow is moving at 104/7 ft/s

or,
let s be the distance of the shadow's tip from the pole. Then when the woman is x ft from the pole, we have
(s-x)/6 = s/13
13(s-x) = 6s
13s-13x = 6s
s = 13/7 x
when x=45, we have
ds/dt = 13/7 * 8 = 104/7 ft/s

To solve this problem, we can use similar triangles and the concept of rates of change.

Let's set up a diagram to visualize the situation:

|
s |
Woman --------------------+---------------------------------- Pole
|
^
|
13 ft
|
v

In the diagram, the woman is represented by the line segment labeled "Woman," the pole is shown as a vertical line, and there is a right triangle formed between the woman, her shadow, and the pole.

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

Since we have similar triangles (the woman, her shadow, and the pole), we can use the property of similar triangles that states the ratios of corresponding sides are equal.

From the diagram, we can write the following equation relating the variables: y / (x + z) = 6 / 13.

We want to find how fast the tip of the shadow is moving, which represents dz/dt. To do that, we need to find an equation that relates x, y, and z.

Differentiating the equation with respect to time (t) gives us: (x + z) * dy/dt = y * dx/dt.

We know that dx/dt is given as 8 ft/s, and we need to find dy/dt when x = 45 ft.

To find y, we need to solve the equation y / (x + z) = 6 / 13 for y.

Dividing both sides of the equation by 6 and then multiplying by 13 gives us:
y = (13 / 6) * (x + z).

Now, we substitute this expression for y into the differentiated equation:
(x + z) * dy/dt = (13 / 6) * (x + z) * dx/dt.

Now we can substitute the known values: x = 45 ft, dx/dt = 8 ft/s into the equation:
(45 + z) * dy/dt = (13 / 6) * (45 + z) * 8.

To find dy/dt, isolate it:
dy/dt = (13 / 6) * (45 + z) * 8 / (45 + z).

Now, we need to find z when x = 45 ft.
Using the equation y / (x + z) = 6 / 13 and substituting the known values, we can solve for z:
y / (45 + z) = 6 / 13.
Cross-multiplying, we get: 13y = 6(45 + z).
Simplifying, we have: 13y = 270 + 6z.
Rearranging the equation: z = (13y - 270) / 6.

Now, we substitute this expression for z into the equation for dy/dt:
dy/dt = (13 / 6) * (45 + (13y - 270) / 6) * 8 / (45 + (13y - 270) / 6).

Finally, substitute the known value of y (which is 6 ft) into the equation to calculate dy/dt:
dy/dt = (13 / 6) * (45 + (13(6) - 270) / 6) * 8 / (45 + (13(6) - 270) / 6).

Simplifying this expression will give us the value of dy/dt, which represents the speed at which the tip of the shadow is moving when the woman is 45 feet from the base of the pole.

To find the speed at which the tip of the woman's shadow is moving when she is 45 feet from the base of the pole, we can use similar triangles and related rates.

Let's define some variables:
- Let h be the height of the woman
- Let x be the distance from the woman to the base of the pole
- Let y be the length of the shadow

We know that the height of the pole is 13 feet and the height of the woman is 6 feet. Therefore, the height of the shadow is given by the proportion:

h/y = (h+13)/x

To find the rate of change of y with respect to x, we can differentiate this equation implicitly with respect to time (t):

d/dt (h/y) = d/dt ((h+13)/x)

To simplify, let's write the above equation as:

h/y = (h+13)/x -----> (1)

Differentiating equation (1) implicitly with respect to t, we get:

d(h/y)/dt = d((h+13)/x)/dt

Differentiating both sides, we have:

(-h/y^2) * dy/dt = (-(h+13)/x^2) * dx/dt

We want to find dy/dt, the rate at which the length of the shadow is changing with respect to time. We are given dx/dt as 8 ft/sec.

Plugging in the given values, we have:

(-6/y^2) * dy/dt = (-(6+13)/45^2) * 8

Simplifying this equation, we obtain:

(-6/y^2) * dy/dt = (-19/2025) * 8

Now, let's solve for dy/dt:

(-6/y^2) * dy/dt = (-19/2025) * 8

Dividing both sides by (-6/y^2) and multiplying by dt, we get:

dy = (-19/2025) * 8 * (y^2/6) * dt

Integrating both sides:

∫dy = ∫(-19/2025) * 8 * (y^2/6) * dt

y = (-19/2025) * 8 * (1/3) * t^3 + C

Since we are given that y = 45 ft when t = 0, we can substitute these values to find the constant C:

45 = (-19/2025) * 8 * (1/3) * 0^3 + C

C = 45

Therefore, the equation for y becomes:

y = (-19/2025) * 8 * (1/3) * t^3 + 45

We want to find dy/dt when y = 45 and x = 45. Let's plug in these values:

dy/dt = d(45)/dt

To find dy/dt, we need to differentiate the equation y = (-19/2025) * 8 * (1/3) * t^3 + 45. Differentiating with respect to t, we get:

dy/dt = (-19/2025) * 8 * (1/3) * 3t^2

Plugging in t = 0, the equation becomes:

dy/dt = (-19/2025) * 8 * (1/3) * 3*0^2
= 0

Therefore, when the woman is 45 feet from the base of the pole, the tip of her shadow is not moving.