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# calculus help please?

word problem help please

### 1 Answer

- LopezLv 54 years agoFavorite Answer
dθ / dt = ( degree for 1 rotation ) / ( period for 1 rotation ) = 2π / 3

By the pic. below, we have

L = 16 * tan θ

dL / dt

= 16 * d tan θ / dt

= 16 * sec^2 θ * ( dθ / dt )

= 16 * sec^2 θ * ( 2π / 3 )

= ( 32π / 3 ) * sec^2 θ

When θ = 15°

dL / dt

= ( 32π / 3 ) * sec^2 15°

= ( 32π / 3 ) * ( 1 + tan^2 15° )

= ( 32π / 3 ) * ( 1 + 7 - 4√3 ) ..... see the note below

= ( 32π / 3 ) * ( 8 - 4√3 ) ..... Ans

Note.

tan^2 15° = 7 - 4√3

pf :

By the formula :

tan ( β / 2 ) = ± √ [ ( 1 - cos β ) / ( 1 + cos β ) ]

So ,

tan^2 ( β / 2 ) = ( 1 - cos β ) / ( 1 + cos β )

Let β = 30° , we have

tan^2 15°

= ( 1 - cos 30° ) / ( 1 + cos 30° )

= [ 1 - ( √3 / 2 ) ] / [ 1 + ( √3 / 2 ) ]

= [ ( 2 - √3 ) / 2 ] / [ ( 2 + √3 ) / 2 ]

= ( 2 - √3 ) / ( 2 + √3 )

= [ ( 2 - √3 )( 2 - √3 ) ] / [ ( 2 + √3 )( 2 - √3 ) ]

= ( 2 - √3 )^2

= 7 - 4√3

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