A street light is at the top of a 14.5 ft. tall pole. A man 5.3 ft tall walks away from the pole with a speed of 5.5 feet/sec along a straight path. How fast is the tip of his shadow moving when he is 47 feet from the pole?

Question

A street light is at the top of a 14.5 ft. tall pole. A man 5.3 ft tall walks away from the pole with a speed of 5.5 feet/sec along a straight path. How fast is the tip of his shadow moving when he is 47 feet from the pole?

Answer 1

I trust you made a sketch.

On mine I let the distance between the man and the post be y ft, and the length of his shadow be x ft
By similar triangles:
5.3/x = 14.5/(x+y)
5.3x + 5.3y = 14.5x
9.2x = 5.3y
92x = 53y
92 dx/dt = 53 dy/dt
but we know dy/dt = 5.5
so dx/dt = 53(5.5)/92 = appr 3.2 ft/sec

Notice that the fact that he is 47 ft from the pole is irrelevant.

Answer 2

the shadow's length is changing at dx/dt.

the tip of the shadow is moving at

dy/dt + dx/dt

Answer 3

Go with Steve's answer.

Mine gives the rate at which the shadow is lengthening, I missed the part where they want the rate at which the shadow is moving.
so
dy/dt + dx/dt
= 5.5 + 3.2
etc

Answer 4

To find the rate at which the tip of the man's shadow is moving, we can use similar triangles and related rates.

Let's define some variables:
h = height of the man (5.3 ft)
x = distance between the man and the base of the pole (47 ft)
y = length of the man's shadow

We are given that the height of the pole is 14.5 ft, so the length of the pole's shadow is h + 14.5.

We need to find the rate at which the tip of the shadow is moving. This is the rate of change of y with respect to time (dy/dt).

Now, let's set up a proportion using the similar triangles formed by the pole, the man, and their respective shadows:

(y + h) / x = 14.5 / y

We can cross-multiply to get:
(y + h) * y = 14.5 * x

Next, we differentiate both sides of the equation with respect to time (t):

d/dt[(y + h) * y] = d/dt(14.5 * x)

Now, let's use the product rule of differentiation:
(dy/dt + dh/dt) * y + (y + h) * dy/dt = 14.5 * dx/dt

We want to find dy/dt, so let's solve for it:
dy/dt = (14.5 * dx/dt - (dy/dt + dh/dt) * y) / (y + h)

Now we can substitute the given values into the equation:
h = 5.3 ft
x = 47 ft
dx/dt = 5.5 ft/sec

Using the given values, we can calculate y:
(y + 5.3) * y = 14.5 * 47
(y^2 + 5.3y) = 14.5 * 47
y^2 + 5.3y - 14.5 * 47 = 0

Solving this quadratic equation will give us the value of y, the length of the man's shadow.

Once we have the value of y, we can substitute it into the dy/dt equation to find the rate at which the tip of the shadow is moving (dy/dt).