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AP Calculus AB· difficulty 3/5

10 ft

A 1010-ft ladder slides down a wall. When the bottom is 66 ft from the wall and moving away at 1 ft/s1~\text{ft/s}, how fast is the top sliding down?

  • A

    1 ft/s1~\text{ft/s}

  • B

    43 ft/s\tfrac{4}{3}~\text{ft/s}

  • C

    34 ft/s\tfrac{3}{4}~\text{ft/s}

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  • D

    12 ft/s\tfrac{1}{2}~\text{ft/s}

Explanation

x2+y2=100x^2 + y^2 = 100. At x=6x = 6: y=8y = 8. Differentiate: 2xx′+2yy′=0⇒y′=−xx′/y=−(6)(1)/8=−3/42x x' + 2y y' = 0 \Rightarrow y' = -x x'/y = -(6)(1)/8 = -3/4. Speed of top is 3/4 ft/s3/4~\text{ft/s} (downward).

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