# A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole?

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To find the rate at which the tip of the man's shadow is moving, we can use related rates. Let ( s ) be the distance between the man and the tip of his shadow, and ( x ) be the distance from the base of the light pole to the man.

( \frac{{ds}}{{dt}} ) = rate of change of distance between man and tip of shadow ( \frac{{dx}}{{dt}} ) = rate of change of distance from base of pole to man

Given: ( \frac{{dx}}{{dt}} = 4 ) ft/s (rate at which the man is walking away from the pole) ( x = 50 ) ft (distance from the base of the pole to the man) Height of pole = 15 ft Height of man = 6 ft

By similar triangles, we have: ( \frac{{s}}{{x}} = \frac{{15}}{{6}} ) ( s = \frac{{5x}}{{2}} )

Differentiate both sides with respect to ( t ): ( \frac{{ds}}{{dt}} = \frac{{5}}{{2}} \frac{{dx}}{{dt}} )

Substitute the given values and solve for ( \frac{{ds}}{{dt}} ): ( \frac{{ds}}{{dt}} = \frac{{5}}{{2}} \times 4 = 10 ) ft/s

Therefore, the tip of the man's shadow is moving at a rate of 10 ft/s when he is 50 ft from the pole.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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