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).