What is the arc length of the curve given by $r (t) = (4 t, 3 t - 6)$ $r(t)=(4t,3t-6)$ in the interval $t \in [0, 7]$ $t in [0,7]$ ?

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What is the arc length of the curve given by $r (t) = (4 t, 3 t - 6)$ $r(t)=(4t,3t-6)$ in the interval $t \in [0, 7]$ $t in [0,7]$ ?

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Answer 1 · 2016-11-25T00:57:26

35

Explanation:

The Arc Length of $f (t) = (x (t), y (t))$ $f(t)=(x(t), y(t) )$ is given by:

$L = \int_{a}^{b} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t$ $L = int_a^bsqrt((dx/dt)^2 + (dy/dt)^2)dt$

Let $x (t) = 4 t \Rightarrow \frac{d x}{d t} = 4$ $x(t)=4t => dx/dt=4$
And, $y (t) = 3 t - 6 \Rightarrow \frac{d y}{d t} = 3$ $y(t)=3t-6=>dy/dt=3$

So,

$L = \int_{0}^{7} \sqrt{{(4)}^{2} + {(3)}^{2}} d t$ $L = int_0^7sqrt((4)^2 + (3)^2)dt$
$:. L = int_0^7sqrt(16+9)dt$
$:. L = int_0^7sqrt(25)dt$
$:. L = 5int_0^7 dt$
$:. L = 5[t]_0^7$
$:. L = 5(7-0)$
$:. L = 35$

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What is the arc length of the curve given by $r (t) = (4 t, 3 t - 6)$ $r(t)=(4t,3t-6)$ in the interval $t \in [0, 7]$ $t in [0,7]$ ?

1 Answer

Explanation:

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1 Answer

Explanation:

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What is the arc length of the curve given by $r (t) = (4 t, 3 t - 6)$ $r(t)=(4t,3t-6)$ in the interval $t \in [0, 7]$ $t in [0,7]$ ?