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

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

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

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

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

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.

To find the position of the particle as a function of time, we need to integrate the given velocity equation with respect to time.

The given velocity equation is:
dx/dt = t^2/2

Integrating both sides of the equation with respect to t, we get:
∫dx = ∫t^2/2 dt

Integrating t^2/2 with respect to t gives us:
x = (1/2) * (∫t^2 dt)

Integrating t^2 gives us:
x = (1/2) * (t^3/3) + C

We still need to find the value of the constant of integration (C). To do this, we use the information given that the particle is at x = 4 when t = 2.

Substituting the values into the equation:
4 = (1/2) * (2^3/3) + C
4 = (1/2) * (8/3) + C
4 = 4/3 + C
C = 4 - 4/3
C = 8/3

Now we have the complete equation for the position of the particle:
x = (1/2) * (t^3/3) + 8/3

Simplifying the equation:
x = t^3/6 + 8/3

Therefore, the correct answer is E. x(t) = t^3/6 - 8/3.

Hmm, let me calculate this for you. But before I do, let me ask you: why did the particle go to therapy?

Because it was feeling "differential"! Okay, now let's solve this differential equation.

To solve dx/dt = t^2/2, we can integrate both sides:

∫dx = ∫(t^2/2)dt

Integrating both sides, we get:

x(t) = (1/2) * (t^3 / 3) + C

Now, to find the constant C, we use the initial condition x = 4 when t = 2:

4 = (1/2) * (2^3 / 3) + C

4 = (8/6) + C

4 = 4/3 + C

C = 12/3 - 4/3

C = 8/3

Therefore, the position of the particle as a function of time is:

x(t) = (1/2) * (t^3 / 3) + 8/3

Simplifying, we have:

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

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

Now, if you excuse me, I'm off to compute the velocity of a clown car. Wish me luck!