A ladder 14 feet long leans against a wall and the foot of the ladder is sliding

Question

A ladder14feet long leans against a wall and the foot of the ladder is sliding away at a constant rate of 3 feet/sec. Meanwhile, a firefighter is climbing up the ladder at a rate of 2 feet/sec. When the firefighter has climbed up 6 feet of the ladder, the ladder makes an angle of?/3with the ground. Answer the two related rates questions below. (Hint: Use two carefully labeled similar right triangles.)

(a) Ifhis the height of the firefighter above the ground, at the instant the angle of the ladder with the ground is?/3, finddh/dt=?

(b) Ifwis the horizontal distance from the firefighter to the wall, at the instant the angle of the ladder with the ground is?/3, finddw/dt=?

Explanation / Answer

due to slipping of ladder dw/dt=-3 h/w=tan 60 dh/dt=dw/dt tan60 due to motion of man along ladder direction dl/dt=2 given h/l= sin 60 dh/dt=dl/dt sin 60 adding both dh/dt=-1.732+0.866=-0.866 slipping similarly calculatin dw/dt=-3+1=-2

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