A winch at the top of a 12-meter building pulls a pipe of the same length to a vertical position, as shown in the figure.

Question

The winch pulls in rope at a rate of −0.1meter per second (ds/dt = −0.1 m/sec).

Find the rate of vertical change and the rate of horizontal change at the end of the pipe when y = 4. (Round your answers to three decimal places.)

vertical

horizontal

Answer 1

x² + y² = 144; (12 - y)² + x² = s². I f y = 4 m, then x² = 144 - 16 = 128; x = √128 = 8√2 m

s² = 128 + 64 = 192; s = √192 = 8√3 m

(12 - y)² + 144 - y² = s²; 144 - 24y + y² + 144 - y² = s²; 288 - 24y = s²; - 24dy/dt = 2sds/dt;

dy/dt = - sds/t/12 = - 8√3(- 0.1)/12 = 0.115 m/sec

2xdx/dt + 2ydy/dt = 0; dx/dt = - (y/x)dy/dt = -(8√3/8√2)·0.11547 m/sec = - 0.141 m/sec