Consider the parametric curve given by the equations:

$\displaystyle y(t) = t^2 +30 t +15$

$\displaystyle x(t) = t^2 +30 t +46$

How many units of distance are covered by the point P(t) = (x(t),y(t)) between t=0, and t=3 ?

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- Dec 10th 2009, 07:38 AMderekjonathonParametric Equation
Consider the parametric curve given by the equations:

$\displaystyle y(t) = t^2 +30 t +15$

$\displaystyle x(t) = t^2 +30 t +46$

How many units of distance are covered by the point P(t) = (x(t),y(t)) between t=0, and t=3 ? - Dec 10th 2009, 07:53 AMTheEmptySet
There are few ways to do this problem

1: use the arc length fromula

$\displaystyle \int_{t_0}^{t_1}\sqrt{(\frac{dx}{dt})^2+(\frac{dy} {dt})^2}dt$

This gives

$\displaystyle \int_{0}^{3}\sqrt{(2t+30)^2+(2t+30)^2}dt$

Just simplify and integrate and you are done.

2: subtract the equations from each other to get

$\displaystyle y-x=-31 \iff y=x-31$

Note that $\displaystyle x(0)=46 \text{ and } x(3)=145$

Since the above is a line with slope one we can use the pythagorean theorem. So both sides have length 99

$\displaystyle c=\sqrt{(99)^2+(99)^2}=99\sqrt{2}$