Answer to Question #350174 in Calculus for HappyFeet

Question #350174

Derive an equation x(t) for the instantaneous position of the car as a function of time. Identify the

●      value x when t = 0 s

●      asymptote of this function as t → ∞


t(0-28 m/s) (s) t(400m)(s) t (Maxspeed)(s)

2.5 9.75 8.0



1
Expert's answer
2022-06-13T16:16:06-0400
"v(t) = A (1-e ^ {- t\/t_{maxspeed}})"


Using the information that "t_{maxspeed}=8.0\\ s" we have



"v(t) = A (1-e ^ {- t\/8.0})"

Using the information that "t (0-28 m\/s)" is "2.5\\ s" we have



"v(2.5) = A (1-e ^ {- 2.5\/8.0})=28""A=\\dfrac{28}{1-e ^ {- 2.5\/8.0}}m\/s"




"A=104.328\\ m\/s"




"v(t) = 104.328 (1-e ^ {- t\/8.0})"

"x(t)=\\int 104.328 (1-e ^ {- t\/8.0})dt"

"=104.328 (t+8e ^ {- t\/8.0})+C"

"x(9.75)=104.328 (9.75+8e ^ {- 9.75\/8.0})+C=400"

"C=-863.912\\ m"

"x(t)=104.328 (t+8e ^ {- t\/8.0})-863.912"


"x(0)=104.328(8)-863.912"

"x(0)=-29.288\\ m"


"x(t)\\to\\infin\\ as\\ t\\to \\infin"

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