An object moves with velocity v(t)=t^2-8t+7
a) write a polynomial expression for the position of the particle at any time t greater or equal to zero.

b) at what time(s) is the particle changing direction

c) find the total distance traveled by the particle from t=0 and t=4

a) write a polynomial expression for the position of the particle at any time t greater or equal to zero. 
Position is found by integrating velocity: 

s(t) = (t^3)/3 - 4t^2 + 7t + c 
where c is a constant corresponding to the position at t=0. 

b) at what time(s) is the particle changing direction
the particle changes direction whenever the velocity is zero; the velocity function equals 

(t-1)(t-7) a difference of squares so the zeros are 1 and 7, it changes direction at 1 second and 7 seconds. 

c) find the total distance traveled by the particle from t=0 and t=4
s(0) = c
s(1) = 8/3 + c
s(4) = 64/3 - 64 + 28 + c. 

from 0 to 1 the particle travels 8/3 units. From 1 to 4 it travels -(64/3 - 36 - 8/3) = (-(56/3 - 108/3)) 
=-(-52/3) = 52/3 units 

so in total it travels 52/3 + 8/3 =20 units

Answer: buoyancy forceexplanation: in simple terms, the principle states that the buoyancy force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational acceleration, g. this is also known as upthrust.