Applications of Differentiation `

用戶92111

2006-11-20 7:00 am

A lighthouse L is built on a small island 3km from the nesrst point A of the straight shoreline .If the lighthouse lamp rotates at 2 revolutions per minute, how fast is the light spot P on the shoreline moving when it is 4km from A?

回答 (1)

Hunter

2006-11-20 7:28 am

✔ 最佳答案

L
|
3km
|
------------4km----------------
A........................................P

Ap = AL * tan (angle ALP)
dAP/dt
= d(AL * tan (angle ALP))/dt
= AL * d (tan (angle ALP))/dt ..............(AL is a constant)
= AL * [sec (angle ALP)]^2 * d(angle ALP)/dt.......(d(angle ALP)/dt is the rotational rate)
Rotational rate
= 2 revolution per min.
= 2 * 2 * pi / 60
=4*pi /60 (radian per second)
[ sec (angle ALP)]^2 =[ tan(angle ALP)]^2 + 1=(4/3)^2 + 1 = 16/9 + 1 = 25/9
The rate of speed
= d(AP)/dt
= 3 * 25/9 * 4 * pi / 60
=1.745km/second.
or
= 104.7km/minute.

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