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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. how fast is the shadow lengthening when she is 35 ft from the base of the pole? (round answers to 3 decimal places.) how fast is the tip of her shadow moving when she is 35 ft from the base of the pole? (round answers to 3 decimal places.)

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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.

User Ravikumar
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