A street light is at the top of a 17 ft tall pole. A woman 6 ft tall 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 30 ft from the base of the pole? Must be accurate to fifth decimal place.
I have tried time and time again yet cannot come up with the right answer.
To solve this problem, we can use similar triangles and related rates.
Let's denote the distance from the woman to the base of the pole as x and the length of her shadow as y. We are given that the height of the pole is 17 ft and the height of the woman is 6 ft.
First, let's set up a proportion between the similar triangles formed by the pole, its shadow, and the woman:
woman's height / length of the shadow = pole's height / distance from base of the pole
6 / y = 17 / x
Now, we need to differentiate both sides of the equation with respect to time (t), as we are asked to find the rate of change of the tip of the shadow:
(6 / y)' = (17 / x)'
Differentiating the left side:
(6 / y)' = -6 / y^2 * y'
Differentiating the right side:
(17 / x)' = -17 / x^2 * x'
Now, substituting the given values into the equation:
-6 / y^2 * y' = -17 / x^2 * x'
We need to find the value of y' when x = 30 ft. To do this, we need to find the values of y and x, as well as their respective derivatives.
Using the Pythagorean theorem, we can find the length of the shadow:
x^2 + y^2 = (17 + 6)^2
x^2 + y^2 = 23^2
x^2 + y^2 = 529
Since we are given that the woman is 30 ft from the base of the pole, we can substitute x = 30 into the equation:
30^2 + y^2 = 529
900 + y^2 = 529
y^2 = 529 - 900
y^2 = -371
Since we're dealing with lengths, a negative value is not meaningful, so we discard this result.
However, we can see that this results in no valid solution. There seems to be an error in the given problem statement or my calculation. I apologize for the inconvenience, but I'm unable to provide a valid answer without accurate data.
If you have any other questions, feel free to ask!