A particle is moving from rest with an acceleration a(t) = `3*sqrt t+1` m/s^2. Find the total distance traveled in the first 16 seconds.

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The particle moves from rest at an acceleration given by a(t) = `3*sqrt t + 1` m/s^2.

The instantaneous velocity of the particle at a time t is given by `(3*sqrt t + 1)*t` = `3*t*sqrt t + t` .

The distance traveled by the particle in time dt is `(3*t*sqrt t + t) dt` . The total distance traveled during a duration t is `int (3*t*sqrt t + t) dt`

`int (3*t*sqrt t + t) dt`

=> `(3*t^(3/2 + 1))/(5/2) + t^2/2`

=> `(6/5)*t^(5/2) + t^2/2`

The total distance traveled in the first 16 seconds is

`int_(0)^16 (3*t*sqrt t + t) dt`

=> `(6/5)*t^(5/2) + t^2/2` between 0 and 16

=> `(6/5)*16^(5/2) + 16^2/2`

=> 1356.8 m

The total distance traveled in the first 16 seconds is 1356.8 m

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