A woman 5 ft tall walks at the rate of 5.5 ft/sec away from a streetlight that is 16 ft above the ground. At what rate is the tip of her shadow moving?

Make the diagram. The streetlight height is 16. Let the distance from the streetlamp base to the tip of the shadow as x. Label the distance from the streetlamp base to the person as d.

Similar triangles:

16/x=5/(x-d)

5x=16x-16d
11x=16d
11 dx/dt=16 dd/dt

dd/dt =5.5ft/sec, solve for dx/dt

I do not agree with bobpursley's solution

let the distance of the woman from the streetlight be x ft
let the length of her shadow be y ft

by ratios ...
5/y = 16/(x+y)
16y = 5x + 5y
11y = 5x
11 dy/dt = 5 dx/dt
but dx/dt = 5.5
dy/dt = (5/11)(5.5) = 2.5

So her shadow is lengthening at 2.5 ft/sec
but she is moving at 5.5 ft/sec, so her shadow is moving at 2.5+5.5 or 8 ft/sec

( I am walking along the inside of a moving train at 3 ft/sec, while the train is moving at 50 ft/sec
So I am moving at 53 ft/sec)

Reiny is definitely correct.

Well, let me shed some light on this question for you with a humorous twist! If a woman walks away from a streetlight at 5.5 ft/sec, we'll call her "The Shadow Seeker." Now, as she moves further away from the streetlight, her shadow is, well, being left behind. It's like the shadow saying, "Hey, don't ghost me!" *wink*

Now, to figure out the rate at which the tip of her shadow is moving, we'll need to do a little math. Since the streetlight is 16 ft above the ground and our protagonist is 5 ft tall, we can imagine a lovely right triangle scenario here. The woman's height forms one side of the triangle, the distance between her and the streetlight forms the other side, and her shadow acts as the hypotenuse.

Using the Pythagorean theorem, we can find the length of the shadow when the woman is at a certain distance from the streetlight. But hold on, we need the rate at which the shadow is moving, not its length. To find that, we'll take the derivative of the shadow's length with respect to time.

So, assuming you're still with me on this comedic math journey, the rate at which the tip of her shadow is moving can be calculated by multiplying the rate at which The Shadow Seeker is moving away from the streetlight (5.5 ft/sec) by the length of her shadow.

Now, since you haven't mentioned any specific distance The Shadow Seeker is from the streetlight, I'll leave you with the equation, and you can plug in the appropriate measurements to calculate the shadow's movement rate:

Rate of tip of shadow movement = (Rate of movement of The Shadow Seeker) * (Length of the shadow)

Remember, my nonsensical friend, don't let your shadow deter you from reaching new heights of laughter and learning!

To find the rate at which the tip of the woman's shadow is moving, we can set up a proportion between the woman's height, her distance from the streetlight, and the length of her shadow.

Let's define the following variables:
- h: height of the woman = 5 ft
- d: distance from the woman to the streetlight (which also represents the length of her shadow)
- s: speed at which the woman is walking away from the streetlight = 5.5 ft/sec
- r: rate at which the tip of the shadow is moving

We need to find the value of r, the rate at which the tip of the shadow is moving.

From the information given, we can use similar triangles to set up the following proportion:

h / d = (h + r) / d

Cross-multiplying the proportion, we get:

h * d = (h + r) * d

Now, let's substitute the given values:
5 ft * d = (5 ft + r) * d

Simplifying the equation, we can cancel out the "d" common on both sides:

5 ft = 5 ft + r

Next, solve for r by isolating the variable:

r = 5 ft - 5 ft

r = 0 ft/sec

Therefore, the rate at which the tip of the woman's shadow is moving is 0 ft/sec. This means that the length of her shadow remains constant since the shadow is not getting longer or shorter as she moves away from the streetlight.