If a street light is at the top of a 12 ft tall pole. a woman 5.5 ft tall walks away from the pole with a speed of 7 ft/sec along a straight path. The tip of the shadow is moving at a rate of approximately 0.0046 ft/sec when the woman is 35 ft from the base of the pole.
What is the rate?
Let x represent the length of the shadow.
Set up the ratios:
12 ft / x = 5.5 ft / (35 ft + x)
Solve for x.
(12 ft) * (35 ft + x) = (5.5 ft) * x
Simplifying the equation:
420 ft + 12x = 5.5x
Subtracting 5.5x from both sides:
6.5x = 420 ft
Dividing both sides by 6.5:
x = 64.615 ft
Rate at which the shadow lengthens.
12 ft / dx = 5.5 ft / (35 ft + x) * dx/dt
Substituting x = 64.615 ft:
12 ft / dx = 5.5 ft / (35 ft + 64.615 ft) * dx/dt
Simplifying:
12 ft / dx = 5.5 ft / 99.615 ft * dx/dt
Cross-multiplying:
12 ft * 99.615 ft * dx/dt = 5.5 ft * dx
Simplifying:
1195.38 ft * dx/dt = 5.5 ft * dx
Dividing both sides by dx:
1195.38 ft * dx/dt = 5.5 ft
Dividing both sides by 1195.38 ft:
dx/dt = 5.5 ft / 1195.38 ft
≈ 0.0046 ft/sec
Therefore the shadow is lengthening at a rate of 0.0046 ft/sec.
Rate at which the tip of the shadow is moving:
dx/dt when x = 64.615 ft:
dx/dt = 0.0046 ft/sec
Therefore the tip of the shadow is moving at a rate of 0.0046 ft/sec when the woman is 35 ft from the base of the pole.