Question

find a vector function, r(t), that represents the curve of intersection of the two surfaces. the paraboloid z = 6x^2 + y^2 and the parabolic cylinder y = 5x^2 r(t) = <t,5t^2,t^2(6 - 25t^2)>

Explanation:

Step1: Parameterize $x$

Let $x = t$.

Step2: Find $y$ in terms of $t$

Since $y = 5x^{2}$ and $x=t$, then $y = 5t^{2}$.

Step3: Find $z$ in terms of $t$

Substitute $x = t$ and $y = 5t^{2}$ into $z=6x^{2}+y^{2}$. So $z = 6t^{2}+(5t^{2})^{2}=6t^{2} + 25t^{4}$.

Answer:

$\mathbf{r}(t)=\langle t,5t^{2},6t^{2}+25t^{4}
angle$