The position function of a spaceship is below and the coordinates of a space station are (8, 5, 11). The captain wants the spaceship to coast into the space station. When should the engines be turned off?
r(t) = (5+t)i + (3+ln(t))j + (9 - 4/(t^2) +1
I noticed that you have unbalanced parentheses in the (9 - 4/(t^2) + 1 part.
To determine when the spaceship should turn off its engines, we need to find the value of t that corresponds to the spaceship reaching the space station coordinates (8, 5, 11).
Given the position function of the spaceship, r(t) = (5+t)i + (3+ln(t))j + (9 - 4/(t^2))k, we can equate the coordinates of the spaceship with the coordinates of the space station, as follows:
x-coordinate: 5 + t = 8
y-coordinate: 3 + ln(t) = 5
z-coordinate: 9 - 4/(t^2) = 11
Solving the first equation for t, we have:
5 + t = 8
t = 8 - 5
t = 3
Substituting t = 3 into the second equation:
3 + ln(t) = 5
ln(3) = 5 - 3
ln(3) = 2
Finally, we substitute t = 3 into the third equation:
9 - 4/(t^2) = 11
9 - 4/(3^2) = 11
9 - 4/9 = 11
81/9 - 4/9 = 11
77/9 = 11
Since 77/9 is not equal to 11, we can conclude that there is no value of t where the spaceship reaches the space station coordinates (8, 5, 11). Therefore, the spaceship cannot coast into the space station and should continue using its engines.