The position function of a spaceship is below and the coordinates of a space station are (8, 6, 10). the captain wants the spaceship to coast into the space station. when should the engines be turned off?
r(t) = (5+t)i + (4+ln(t))j + (8-(4/t6+1))k
t = ( )

at time t= 1

Step-by-step explanation: shown in the attachment

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.