题目

一个微积分的题
A lighthouse is fixed 100 feet from a straight shoreline.A spotlight revolves at a rate of 16 revolutions per minute,(32 rad/min ),shining a spot along the shoreline as it spins.At what rate is the spot moving when it is along the shoreline 16 feet from the shoreline point closest to the lighthouse?

提问时间：2021-03-19

答案

Solution:
Let O be the position of lighthouse.
Let A be the point on the shoreline closest to the lighthouse.
Let P be the point on the shoreline spotted by the lighthouse at a moment.
Let r be the distance between O and P.
Let θ be the angle between OA and OP.
∵Angular speed = 32π/min
∴Tangetial speed at point P = 32πr
∴Linear speed of point P moving along the shoreline
= 32πr/cosθ
= 32πr/(100/r)
= 8πr²/25
= 8π(100²+16²)/25
= 82048π/25
≈ 10310.46 (feet/min)