given that ( s(t)=12 t - t^{2} ) is a position function where ( s(t) ) is measured in millimeters and ( t )…

given that ( s(t)=12 t - t^{2} ) is a position function where ( s(t) ) is measured in millimeters and ( t ) is time in seconds, find the average velocity (average rate of change) for ( s(t) ) between ( t = 1 ) and ( t = 3 ) seconds. *algebraic work must be shown to retain points on quizzes, midterm or final

given that ( s(t)=12 t - t^{2} ) is a position function where ( s(t) ) is measured in millimeters and ( t ) is time in seconds, find the average velocity (average rate of change) for ( s(t) ) between ( t = 1 ) and ( t = 3 ) seconds. *algebraic work must be shown to retain points on quizzes, midterm or final

Answer

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). For the position function (s(t)=12t - t^{2}) and the interval ([1,3]), we have (a = 1), (b = 3).

Step2: Calculate (s(3)) and (s(1))

  • Calculate (s(3)): Substitute (t = 3) into (s(t)): (s(3)=12\times3-3^{2}=36 - 9=27)
  • Calculate (s(1)): Substitute (t = 1) into (s(t)): (s(1)=12\times1-1^{2}=12 - 1 = 11)

Step3: Apply the average - rate - of - change formula

(\frac{s(3)-s(1)}{3 - 1}=\frac{27-11}{2}=\frac{16}{2}=8)

Answer:

(8) millimeters per second