What is the average speed of an object that is not moving at $t=0$ and accelerates at a rate of $a(t) =8-t^2/4$ on $t in [2,5]$ ?

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Answer 1 · 2016-06-17T09:23:39

$v_a=28.67" "("unit")/s$

" $"we can calculate using the average value theorem"$
$v_a=1/(5-2)int _2^5 v(t)*d t$

$"v(t) represents the speed function with respect to time"$

$"first we must find the function v(t)"$

$v(t)=int a(t)*d t; " so " a(t)=8-t^2/4$

$v(t)=int (8-t^2/4)d t$

$v(t)=8t-1/4*1/3*t^3+C$

$"C=0 because it isn't moving at t=0"$

$v(t)=8t-1/12t^3+0$

$"now we can calculate the average speed of the object"$

$v_a=1/(5-2)int_2^5 (8t-1/12*t^3)d t$

$v_a=1/3[|8*1/2*t^2-1/12*1/4*t^4|_2^5]$

$v_a=1/3[|4t^2-1/48t^4|_2^5]$

$v_a=1/3[(4*5^2-1/48*5^4)-(4*2^2-1/48*2^4)]$

$v_a=1/3[(100-625/48)-(16-16/48)]$

$v_a=1/3[(4800-625)/48-47/48]$

$v_a=1/3[(4175-47)/48]$

$v_a=1/3[4128/48]$

$v_a=4128/144$

$v_a=28.67" "("unit")/s$

What is the average speed of an object that is not moving at t=0t=0 and accelerates at a rate of a(t) =8-t^2/4a(t) =8-t^2/4 on t in [2,5]t in [2,5]?