Question #148530

A ladder 15ft long rest against a vertical wall. Its top slides down the wall while its bottom moves away along with the level ground at a speed of 2 ft/s . How fast is the angle between the top of the ladder and the wall changing when the angle is π/3 radians.

Expert's answer

Ladder and wall can be regarded as sides of a right triangle

"At t = 0, OB=0 and OC =BC =15 ft. OB =2t ft"

"sin \\theta=\\frac{OB}{BC} \\implies \\theta = sin^{-1}\\frac{OB}{BC}"

"\\theta = sin^{-1}\\frac{2t}{15} \\implies d\\theta=\\frac{2}{\\sqrt{225-4t^2}}dt"

"\\frac{d\\theta}{dt}=\\frac{2}{\\sqrt{225-4t^2}}"

Determining t, when "\\theta=\\frac{\\pi}{3}"

"sin\\frac{\\pi}{3}=\\frac{2t_1}{15} \\implies t_1= \\frac{15\\sqrt{3}}{4}"

"\\frac{d\\theta}{dt}_{t=t_1}=\\frac{2}{\\sqrt{225-4t^2}}=\\frac{2}{\\sqrt{225-4(\\frac{15 \\sqrt{3}}{4})^2}}"

"=\\frac{2}{\\sqrt{225-225(\\frac{{3}}{4})}}=\\frac{4}{15}" rad/s

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