Weight on a spring moves so that it's speed is given by the formula.. Y=10+2sin((pie x t)/ 6) What is the greatest speed that the weight can reach?

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Given spped

`Y=10+2sin((pi t)/ 6)` (i)

diiferentiate with respect to t

`(dY)/(dt)=((2pi)/6)cos((pit)/6)`

`` For maximum speed ,

`(dY)/(dx)=0`

`cos((pit)/6)=0`

`cos((pit)/6)=cos(pi/2)`

`(pit)/6=pi/2`

`t=3` sec

`(d^2Y)/(dt^2)=-(pi^2/18)sin((pit)/6)`

`(d^2Y)/(dt^2)}_{t=3}=-(pi^2/18)<0`

`Y }_{t=3)=10+2sin((3pi)/6)=10+2=12` unit/sec

`y=10+2sin((pit)/6)`

`y'=pi/3cos((pit)/6)`

`y'=0 rArr` `(pit)/6=pi/2` `(pit)/6=(3pi)/2` `t_1=3` `t_2=9`

`y''=-1/2(pi/3)^2sin((pit)/6)`

`y''(3)=-1/2(pi/3)^2sin(pi/2)=-1/2(pi/3)^2<0`

``Max

`y''(9)=-1/2(pi/3)^2sin((3pi)/2)=1/2(pi/3)^2`

Min

Max value:

`y(3)=10+2sin(pi/2)=12`

Min value:

`y(9)=10+2sin((3pi)/2)` `=8`

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