`x=3t+5 , y=7-2t , -1<=t<=3` Find the arc length of the curve on the given interval.

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The formula of arc length of a parametric equation on the interval `alt=tlt=b` is:

`L = int_a^b sqrt((dx/dt)^2+(dy/dt)^2) dt`

The given parametric equation is:

`x=3t + 5`

`y=7 - 2t`

The derivative of x and y are:

`dx/dt = 3`

`dy/dt = -2`

So the integral needed to compute the...

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The formula of arc length of a parametric equation on the interval `alt=tlt=b` is:

`L = int_a^b sqrt((dx/dt)^2+(dy/dt)^2) dt`

The given parametric equation is:

`x=3t + 5`

`y=7 - 2t`

The derivative of x and y are:

`dx/dt = 3`

`dy/dt = -2`

So the integral needed to compute the arc length of the given parametric equation on the interval `-1lt=tlt=3` is:

`L = int_(-1)^3 sqrt(3^2+(-2)^2) dt`

The simplified form of the integral is:

`L = int_(-1)^3 sqrt13 dt`

Evaluating this yields:

`L = sqrt13t `  `|_(-1)^3`

`L = sqrt(13)*3 - sqrt13*(-1)`

`L=3sqrt13 + sqrt13`

`L=4sqrt13`

Therefore, the arc length of the curve is `4sqrt13` units.

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