$\text { Find the arc length parameter along the curve from the point where } t = 0 \text { by evaluating } s = \int _ { 0 } ^ { t } | v ( \tau ) | d \tau \text {. }$
-r(t) = (1 + 3t)i + (1 + 6t)j + (4 - 4t)k
A) $5 \mathrm { t }$
B) $2 \sqrt { 13 } t$
C) $3 \sqrt { 5 } t$
D) $\sqrt { 61 } \mathrm { t }$

Question

$\text { Find the arc length parameter along the curve from the point where } t = 0 \text { by evaluating } s = \int _ { 0 } ^ { t } | v ( \tau ) | d \tau \text {. }$
-r(t) = (1 + 3t)i + (1 + 6t)j + (4 - 4t)k 
A) $5 \mathrm { t }$
B) $2 \sqrt { 13 } t$
C) $3 \sqrt { 5 } t$
D) $\sqrt { 61 } \mathrm { t }$

Quizplus · Accepted Answer

The velocity vector $ v(t) $ is the derivative of $ r(t) $, which is $ (3, 6, -4) $. The magnitude of $ v(t) $ is $\sqrt{3^2 + 6^2 + (-4)^2} = \sqrt{61}$. Thus, the arc length parameter $ s = \int_{0}^{t} \sqrt{61} \, d\tau = \sqrt{61} \, t $.

-R(t) = (1 + 3t)i + (1 + 6t)j 5t5 \mathrm { t }5t

-R(t) = (1 + 3t)i + (1 + 6t)j $5 \mathrm { t }$