To determine the position of the rocket ship after 10 seconds, we will use the given parametric equations [tex]\( x(t) = 3t \)[/tex] and [tex]\( y(t) = 4t^2 + 1 \)[/tex].
1. Calculate the x-coordinate:
The parametric equation for the x-coordinate is given by [tex]\( x(t) = 3t \)[/tex].
Substitute [tex]\( t = 10 \)[/tex] seconds into the equation:
[tex]\[
x(10) = 3 \cdot 10 = 30
\][/tex]
2. Calculate the y-coordinate:
The parametric equation for the y-coordinate is given by [tex]\( y(t) = 4t^2 + 1 \)[/tex].
Substitute [tex]\( t = 10 \)[/tex] seconds into the equation:
[tex]\[
y(10) = 4 \cdot (10)^2 + 1 = 4 \cdot 100 + 1 = 400 + 1 = 401
\][/tex]
3. Determine the coordinates of the ship:
After 10 seconds, the coordinates of the ship are [tex]\( (x(10), y(10)) \)[/tex]:
[tex]\[
(x(10), y(10)) = (30, 401)
\][/tex]
Therefore, the position of the rocket ship after 10 seconds is [tex]\( (30, 401) \)[/tex].
The correct answer is:
[tex]\[
(30, 401)
\][/tex]
