Distance traveled over an interval of time  can be found by integrating the velocity function over this interval:

For the velocity function

Over the interval of time , the distance traveled is

Distance traveled over an interval of time  can be found by integrating the velocity function over this interval with respect to time:

For the velocity function

The integral of a constant  is simpy , and the integral of an exponential function  is .

We'll treat the ending time as unknown and write a general distance equation:

The time when he crosses the finish line is when this equation is equal to :

This occurs at time

Begin by finding the velocity function, which is the integral of the acceleration function with respect to time.

For acceleration

The integral of a constant  is simpy , and the integral of an exponential function  is .

The velocity function can be found using these properties to be

The constant of integration can be found utilizing the initial condition:

This gives the complete function

Now to find distance:

Distance traveled over an interval of time  can be found by integrating the velocity function over this interval:

Over the inteval 

Note that the integral of a function  is . 

Distance traveled over an interval of time  can be found by integrating velocity over this interval:

For the velocity function

We do not know the end times, but we can assume that the timer starts at zero. Therefore, we can write a distance equation of the form:

The integral of a constant  is simpy , and the integral of a function of the form  is ., where 

Now, begin by finding the time it takes Lacey to run fourteen feet:

So there are two roots ; however, a negative time is not possible, so she runs the first fourteen feet in two seconds.

Now, calculate the time it takes her to run twenty-eight feet:

The positive root is 

So the time it takes Lacey to run the additional fourteen feet after the first fourteen feet is the difference of these two times:

From the position equation 

we see that time is in the unit of seconds.  Therefore, we must convert  into .  Since there are  in  and  in , the total amount of time in seconds in .  Plugging into the position equation gives us .  After rounding the answer, we get  or 

.  

From the passage, the position the ball travels across the Mississippi River is

From this equation, we let  and plug into the equation to get the distance traveled () 

We have to distinguish distance from displacement. If we wanted to find the displacement of the object, we would take the definite integral of velocity and solve. However, when we solve for distance, we have to solve for the integral of 

We take the absolute value because distances can never be negative. 

Taking the integral of an absolute value function is difficult, instead we will do separate integrals for when  is positive and negative between  and .

  from  to . 

 from  to 

We can express the distance travelled  as

We split it up into two integrals because the sign of the integral would change, which is compensated for by the absolute value. 

Firstly, recall that

Also recall that

Knowing these two identites, we can solve for 

Solving for ,

There are two things we will need to solve for.  First is the vertex of the parabola, and second is where the ball will hit the ground.

Use the FOIL method to simplify the binomials of the height function.

Recall the vertex formula for the parabola to find the max.  Substitute the values of  and  since the height function is in standard form.

This is the position of the ball that is thrown from the skyscraper.

We will now need to know the roots of the function.  This is where the ball hits the ground.  Set the height function equal to zero and solve for the roots.

Since there is no such thing as negative distance, we will only consider the positive distance .  This is where the ball has hit the ground.

Subtract  from where the ball has been thrown from  where the ball has hit the ground.  This is the horizontal distance the ball has been thrown.

To solve, simply realize you were given d(t) which is the distance function. Thus, just plug in .

We know that distance is the difference of  positions, so we need a position function. We can find this by integrating the velocity function.

Recall to integrate this particular function we use, 

.

The car started at position , so , and we find

and . Our position function is now determined.

The distance the car traveled between  and  is the difference between the positions of the car at those  times. Hence, we have .