A street light is at the top of a 13 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 35 ft from the base of the pole?
Let the woman be x feet from the pole, and the shadow tip be s feet in front of the woman.
By similar triangles,
s/6 = (x+s)/13
s = 6/7 x
ds/dt = 6/7 dx/dt = 6/7 * 7 = 6
The shadow is moving at 6ft/s from the woman, and she's moving 7ft/s from the pole.
so, the shadow moves at 13ft/s regardless of the distance from the pole.
To solve this problem, we can use similar triangles and the chain rule from calculus.
Let's denote:
h = height of the pole = 13 ft
x = distance of the woman from the base of the pole
y = length of the shadow
Based on the similar triangles, we have the following proportions:
y/h = (y + 6)/(x)
To find the rate of change of the shadow length with respect to time, we need to differentiate both sides of the equation with respect to time:
dy/dt / h = (6 * dx/dt) / (x^2)
Now, we can substitute the given values into the equation:
h = 13 ft
dx/dt = 7 ft/sec
x = 35 ft
Plugging in these values, we get:
dy/dt / 13 = (6 * 7) / (35^2)
Simplifying further, we have:
dy/dt = (6 * 7 * 13) / (35^2)
Calculating this expression, we find:
dy/dt = 3.36 ft/sec
Therefore, the tip of her shadow is moving at a rate of 3.36 ft/sec when she is 35 ft from the base of the pole.
To find the rate at which the tip of the woman's shadow is moving, we need to use similar triangles and related rates.
Let's denote the height of the woman as "h" and the distance between the woman and the base of the pole as "x". We are given that h = 6 ft and x is changing at a rate of dx/dt = 7 ft/sec.
Let's also call the length of the shadow "s" and the height of the street light "y". We are given that y = 13 ft.
From the similar triangles formed by the woman, her shadow, and the pole, we can write:
h / x = (h + y) / s
Rearranging this equation, we get:
s = (h + y) * (x / h)
Now, to find how fast the shadow is changing, we need to differentiate s with respect to time (t).
ds/dt = [(h + y) * (dx/dt) * h - x * (dh/dt) * (h + y)] / h^2
Since dh/dt represents the rate at which the height of the woman is changing, and it is not given, we can assume the woman's height is not changing.
Therefore, dh/dt = 0.
Substituting the given values, we have:
ds/dt = [(6 + 13) * (7) * 6 - (35) * (0) * (6 + 13)] / 6^2
Simplifying this equation, we get:
ds/dt = 19 ft/sec
Therefore, the tip of the woman's shadow is moving at a rate of 19 ft/sec when she is 35 ft from the base of the pole.