Explanation Formula used: Let r ( t ) be a smooth curve. If | v ( t ) | = 1 , then the curve uses arc length as a parameter. The arc length parameterization of the curve is s ( t ) = ∫ a t | v ( u ) | d u , where | v | = | r ′ | . Calculation: Compute r ′ ( t ) . r ′ ( t ) = 〈 ( 17 cos t ) ′ , ( 15 sin t ) ′ , ( 8 sin t ) ′ 〉 = 〈 − 17 sin t , 15 cos t , 8 cos t 〉 Thus, the component of r ′ ( t ) is 〈 − 17 sin t , 15 cos t , 8 cos t 〉 . Compute | v ( t ) | by taking magnitude of r ′ ( t ) . | v ( t ) | = | 〈 − 17 sin t , 15 cos t , 8 cos t 〉 | = ( − 17 sin t ) 2 + ( 15 cos t ) 2 + ( 8 cos t ) 2 = 289 sin 2 t + 225 cos 2 t + 64 cos 2 t = 289 sin 2 t + 289 cos 2 t On further simplification, the value of | v ( t ) | is obtained as follows. | v ( t ) | = 289 ( sin 2 t + cos 2 t ) = 289 ( 1 ) = 289 = 17 Thus, the value of | v ( t ) | ≠ 1 implies that the curve r ( t ) doesn’t use arc length as a parameter. Find a description that uses arc length as a parameter. Substitute t = s 17 in r ( t ) , r ( t ) = 〈 17 cos ( s 17 ) , 15 sin ( s 17 ) , 8 sin ( s 17 ) 〉 , for 0 ≤ t ≤ 17 π Compute r ′ ( s )

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Chapter 14.4, Problem 38E

To determine

Whether the curve r(t)=〈17cost,15sint,8sint〉, for 0≤t≤π uses arc length as a parameter or not and find a description that uses arc length as a parameter if not.

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Chapter 14.4, Problem 37E

Chapter 14.4, Problem 39E

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Chapter 14 Solutions

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Whether the curve r ( t ) = 〈 17 cos t , 15 sin t , 8 sin t 〉 , for 0 ≤ t ≤ π uses arc length as a parameter or not and find a description that uses arc length as a parameter if not.

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Whether the curve r(t)=〈17cost,15sint,8sint〉, for 0≤t≤π uses arc length as a parameter or not and find a description that uses arc length as a parameter if not.

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