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#