Question

At what point do the curves r1(t) = t, 5 − t, 48 t2 and r2(s) = 8 − s, s − 3, s2 intersect?

Answer 1

Set components equal:
$t = 8-s \\ 5-t = s-3 \\ 48 t^2 = s^2$

From this we find:
$t = 8-s \\ s = 4 \sqrt{3} t$

Substitute and solve system:
$t = \frac{8}{1+4 \sqrt{3}} = 1.009 \\ \\ s = 8-t = 8 - 1.009 = 6.991$

The point where they intersect is:
$(1.009, 5-1.009, 48(1.009)^2)$