$\text { Find } \mathbf { r } ( t ) \text { given the following. }$ $\mathbf { r } ^ { t } ( t ) = 18 t ^ { 5 } \mathbf { j } + 6 t \mathbf { k } , \mathbf { r } ( 0 ) = 2 \mathbf { i } + 18 \mathbf { j }$
A) $\mathbf { r } ( t ) = 2 \mathbf { i } + \left( 18 + 3 t ^ { 6 } \right) \mathbf { j } + 3 t ^ { 2 } \mathbf { k }$
B) $\mathbf { r } ( t ) = 2 \mathbf { i } + 3 t ^ { 2 } \mathbf { j } + \left( 18 + 3 t ^ { 6 } \right) \mathbf { k }$
C) $\mathbf { r } ( t ) = \left( 18 + 3 t ^ { 6 } \right) \mathbf { j } + 3 t ^ { 2 } \mathbf { k }$
D) $\mathbf { r } ( t ) = 2 \mathbf { i } + \left( 18 - 3 t ^ { 6 } \right) \mathbf { j } - 3 t ^ { 2 } \mathbf { k }$
E) $\mathbf { r } ( t ) = 18 \mathbf { i } + \left( 2 + 3 t ^ { 6 } \right) \mathbf { j } + 3 t ^ { 2 } \mathbf { k }$

Question

Quizplus · Accepted Answer

To find $\mathbf { r } ( t )$, integrate each component of $\mathbf { r } ^ { t } ( t )$. Integrating $18t^5$ with respect to $t$ gives $3t^6$, and integrating $6t$ gives $3t^2$. The constants of integration are determined by the initial condition $\mathbf { r } ( 0 ) = 2 \mathbf { i } + 18 \mathbf { j }$, leading to the correct expression for $\mathbf { r } ( t )$.

Find r(t) given the following. \text { Find } \mathbf { r } ( t ) \text { given the following. } Find r(t) given the following.

$\text { Find } \mathbf { r } ( t ) \text { given the following. }$