We can simply plug in the values into our first equation by:

By the fundamental theorem of calculus and since : 

By the given formula:

By the fundamental theorem of calculus and because : 

The velocity of an object is defined as the derivative of its position with respect to time; in turn, the acceleration of an object is defined as the derivative of its velocity with respect to time. Since we've been given the equation defining the particle's position after  seconds, to determine its acceleration we must take the second derivative of this equation.

The first derivative with respect to time is

.

The second derivative with respect to time is

.

Now we simply substitute  for the new equation for the particle's acceleration to yield

We can model the flow of water into the pool by a function of time; let  be the amount of water in the pool at time . Then  represents the rate at which water flows into the pool at time . By the net change theorem, evaluating the definite integral of  over an interval  yields the total change, or amount of, that quantity over that interval. Hence, we can determine how much water flowed into the pool over the first  minutes by evaluating the definite integral of  from  to , as follows:

Hence, the amount of water that flows into the pool during the first  minutes at this rate is .

In order to figure out the particle's position at time , we must first discover the position as a function of time.

We know that

,

and that the velocity at time is given by 

.

Since the initial position is not known, we might as well say that the initial position is at the origin, that is to say that

.

Then will tell us how far the particle has traveled from the origin, which would be exactly the distance the particle has traveled from it's original position. First, we must compute . We know that, 

so then

,

where is some constant. To find , we use the condition that . Plugging this in to the velocity function, we get

.

Thus our velocity as a function of time is

.

Now to find position, we know that

.

Plugging in , we get

,

where again is some unknown constant. To solve for , we plug in the condition and obtain .

Thus our position as a function of time is given by .

Now it is just a matter of plugging in . Doing so, we obtain .

Therefore our distance away from where the particle started is .

The position of a particle can be found by integrating the velocity function.  We need to use the initial condition to find our constant, since it will be an indefinite integral.

Therefore, our constant equals 

Putting it all together, our equation is:

First, integrate the velocity function because the integral will be the position function: . Then, integrate. Remember to raise the exponent by 1 and then put that result on the denominator: . Then, plug in your initial conditions to find your C: . Simplify and solve for C. . Plug that in to get your position function: .

First, write an integral expression. Remember that the integral of velocity is position:

Then, integrate. Remember to raise the exponent by 1 and then also put that result on the denominator. Also, add a C because it is an indefinite integral at this point:

To find C, plug in your initial conditions:

Solve for C: 

Plug in to get your position function:

First, write an integral expression. Remember that the integral of the velocity function is the position function:

Integrate. Remember to add one to the exponent and then also put that result on the denominator. Also, remember to add C because it is an indefinite integral at this point:

Now, plug in your initial conditions to find C:

Now, solve for C:

Sub C into your position function: