Find the Length of the Indicated Portion of the Curve $\mathbf { r } ( t ) = ( 4 \cos t ) \mathbf { i } + ( 4 \sin t ) \mathbf { j } + 5 t \mathbf { k } , 0 \leq t \leq \pi / 2$

question 30

Multiple Choice

Find the length of the indicated portion of the curve.
- $\mathbf { r } ( t ) = ( 4 \cos t ) \mathbf { i } + ( 4 \sin t ) \mathbf { j } + 5 t \mathbf { k } , 0 \leq t \leq \pi / 2$

• A) \(\frac { 3 t } { 2 } \pi\)
• B) \(\frac { \sqrt { 41 } t } { 2 } \pi\)
• C) \(\frac { \sqrt { 57 } \mathrm { t } } { 2 } \pi\)
• D) \(\frac { \sqrt { 41 } t } { 4 } \pi\)

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Definitions:

Survival

The ability of an organism to continue living or existing, often despite facing adverse conditions.

Exponential Growth

Growth whose rate becomes ever more rapid in proportion to the growing total number or size.

Arithmetic Growth

A pattern of growth where a population increases by a constant amount in each time period.

Logistic Growth

Population increase that results in an S-shaped curve; growth is slow at first, steepens, and then levels off due to environmental resistance.