The velocity of a particle on the x-axis is given by the differential equation dx/

Question

dt= t^2/ 2 and the particle is at x = 4 when t = 2. The position of the particle as a function of time is:

A. x(t) = t^3 / 6 - 26/3
B. x(t) =t +2
C. r(t) = t - 2
D. x(t) = t^3 / 2
E. x(t) = t^3 / 6 - 8/3

Answer 1

dx/dt = t^2/2
dx = t^2/2 dt
x = 1/6 t^3 + C
so already A and E are the only choices.
now use x(2) = 4 to find what C is

Answer 2

E. x(t) = t^3 / 6 - 8/3

Answer 3

E. x(t) = t^3 / 6 + 8/3

Answer 4

Let's substitute the value of t = 2 into the equation x = 1/6 t^3 + C:

x(2) = 1/6 * (2)^3 + C
4 = 1/6 * 8 + C
4 = 4/6 + C
4 - 4/6 = C
24/6 - 4/6 = C
20/6 = C
10/3 = C

Now we can write the equation for x(t) with the value of C:

x(t) = 1/6 t^3 + 10/3

So the correct answer is E. x(t) = t^3 / 6 + 8/3.