Questions: Does the point P=(7,37,1296) lie on the path r=<1+t, 2+t^2, t^4> ? No, it doesn't. Yes, it does. Does the point Q=(-1,7,16) lie on the path r=<1+t, 2+t^2, t^4> ? Yes, it does. No, it doesn't.

Does the point P=(7,37,1296) lie on the path r=<1+t, 2+t^2, t^4> ?
No, it doesn't.
Yes, it does.

Does the point Q=(-1,7,16) lie on the path r=<1+t, 2+t^2, t^4> ?
Yes, it does.
No, it doesn't.

Transcript text: Does the point $P=(7,37,1296)$ lie on the path $\mathbf{r}=\left\langle 1+t, 2+t^{2}, t^{4}\right\rangle$ ? No, it doesn't. Yes, it does. Does the point $Q=(-1,7,16)$ lie on the path $\mathbf{r}=\left\langle 1+t, 2+t^{2}, t^{4}\right\rangle$ ? Yes, it does. No, it doesn't.

Solution

Solution Steps

To determine if a point lies on the given path, we need to find a value of $ t $ such that the parametric equations of the path match the coordinates of the point. Specifically, for a point $ (x, y, z) $, we need to solve the system of equations: $ 1 + t = x $, $ 2 + t^2 = y $, and $ t^4 = z $. If a common $ t $ satisfies all three equations, the point lies on the path.

Step 1: Check Point $ P = (7, 37, 1296) $

To determine if point $ P $ lies on the path defined by $ \mathbf{r} = \langle 1+t, 2+t^2, t^4 \rangle $, we set up the following equations based on the coordinates of $ P $:

$ 1 + t = 7 $
$ 2 + t^2 = 37 $
$ t^4 = 1296 $

Solving these equations, we find:

From $ 1 + t = 7 $, we get $ t = 6 $.
Substituting $ t = 6 $ into $ 2 + t^2 = 37 $ gives $ 2 + 36 = 38 $, which does not hold.
Therefore, point $ P $ does not satisfy all equations.

Step 2: Check Point $ Q = (-1, 7, 16) $

Next, we check if point $ Q $ lies on the same path. We set up the equations:

$ 1 + t = -1 $
$ 2 + t^2 = 7 $
$ t^4 = 16 $

Solving these equations, we find:

From $ 1 + t = -1 $, we get $ t = -2 $.
Substituting $ t = -2 $ into $ 2 + t^2 = 7 $ gives $ 2 + 4 = 6 $, which does not hold.
Therefore, point $ Q $ does not satisfy all equations.