A street light is at the top of a 19 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 4 ft/sec along a straight path. How fast is the very tip of her shadow moving when she is 35 feet from the base of the pole?
To solve this problem, we can use similar triangles. Let's say the distance from the base of the pole to the tip of the woman's shadow is x.
According to the problem, the height of the pole is 19 feet, and the height of the woman is 6 feet. The height of the shadow can be represented as (19 - 6) = 13 feet.
Since the triangles formed by the pole and the shadow are similar, we can write the following proportion: (height of the pole) / (distance from the base of the pole to the tip of the woman's shadow) = (height of the shadow) / (distance from the base of the pole to the tip of the actual shadow)
19 / x = 13 / (x + 35)
To find dx/dt, the rate at which the tip of the shadow moves, we differentiate both sides of the equation with respect to time (t):
(d/dt)(19 / x) = (d/dt)(13 / (x + 35))
Using the quotient rule, we get:
-19(x^(-2))(dx/dt) = -13(x + 35)^(-2)(dx/dt)
Simplifying:
-19 / x^2 = -13 / (x + 35)^2
Cross-multiplying:
13x^2 = 19(x + 35)^2
Expanding and rearranging:
13x^2 = 19(x^2 + 70x + 1225)
13x^2 = 19x^2 + 1330x + 23175
6x^2 - 1330x - 23175 = 0
Using the quadratic formula:
x = (-(-1330) ± sqrt((-1330)^2 - 4 * 6 * (-23175))) / (2 * 6)
Solving for x, we get two possible values:
x ≈ 60.05 or x ≈ -64.72
Since distance cannot be negative, we discard the negative solution. Therefore, x ≈ 60.05 feet.
To find the rate at which the tip of the shadow is moving, dx/dt, we substitute x = 60.05 into the first equation we derived:
19 / x = 13 / (x + 35)
19 / 60.05 = 13 / (60.05 + 35)
Simplifying:
0.3161 ≈ 0.1382 / (95.05)
Cross-multiplying:
0.3161 * 95.05 = 0.1382
Approximately:
30.08 = 0.1382
Converting feet per second to inches per second, we get:
dx/dt ≈ 30.08 * 12 ≈ 360.96 in/sec
So, the tip of the woman's shadow is moving at a rate of approximately 360.96 inches per second when she is 35 feet from the base of the pole.