You are a competitive bicyclist. During a race, you bike at a constant velocity of k meters per second. A chase car waits for you at the ten- mile mark of a course. When you cross the ten-mile mark, the car immediately accelerates to catch you. The position function of the chase car is given by the equation s(t)=15/4t^2- 5/12t^3, for 0<t<6, where t is the time in seconds and s is the distance traveled in meters. When the car catches you, you and the car are traveling at the same velocity, and the driver hands you a cup of water while you continue to bike at k meters per second.

Part 2

2. Use your answer to Exercise 1 and the given information to write an equation that represents the velocity k at which the chase car catches you in terms of t.

k(t)= t(15/4t- 5/12t^2)

To get answer all they wanted was for you to take the derivative of the function.

Part 3

Find the velocity function of the car.

v(t)= 15/2t- 5/4t^2

V=Velocity

T=time

To get solution, you have to take the derivative again (Second Deritative) of the function you found in Part 2

Use your answers to Exercises 2 and 3 to find how many seconds it takes the chase car to catch you.

Set

K= 15/4t- 5/12t^2 and

V(t)= 15/2 t- 5/4t^2 to each other

15/2t-5/4t^2=15/4t-5/12t^2

and solve for t to get t=4.5 seconds.

Part 1

Write an equation that represents your position s (in meters at time) at time t (in seconds)

Awnser= s=kt

s=distance

k=Velocity

t= Time in seconds

Velocity/ Time= Distance

**Chapter 2 Project**

Part 4

Part 5

What is your velocity when the car catches you?

s=(kt)

plug in 4.5 so

s=K(4.5)= K(4.5)= 15/4(4.5) - 5/12 (4.5)^2

=8.4375

Part 6

6. Use a graphing ability to graph the chase car's position function in the same viewing window.

Part 7

Find the point of intersection of the two graphs in exercise 6. What does this point represent in the context of the problem?

T=4.5

Point of intersection (4.5, 37.96875)

To find point of intersection, use the intersect option on your graphing calculator. In the context of the problem, it shows the time that the car and bike are traveling in the same velocity, making the hand off possible.

Part 8

Describe the graphs in Exercise 6 at the point of intersection. Why does this point represent in the context of the problem?

Part 9

Suppose you bike at a constant velocity of 9 meters per second and the chase car's position function is unchanged.

a- Use a graphing utility to graph the chase car's position function and your position function in the same viewing window.

b- In this scenario, how many times will the chase car be in the same position as you after the 10 mile mark?

c- In this scenario, would the driver of the car be able to successfully hand off a cup of water to you? Explain.

In the context of the problem, it represents the time in which the hand off of water takes place because both bike and car are traveling at the same velocity.

Part 9

s(t)=15/4t^2-5/12t^3

s=9t

If you were to graph the functions in a calculator, you would see that the two never touch each other, meaning that at no point in time will the bike and car ever have the same position, therefore a handoff is impossible

Part 10

Suppose you bike at a constant velocity of 8 meters per second and the chase car's position function is unchanged.

a- Use a graphing utility to graph the chase car's position function and your position function in the viewing window.

b- In tis scenario, how many times wil the chase car be in the same position as you after the 10 mile mark.

c-In this scenario, why might it be difficult for the driver of the chase car to successfully handoff a cup of water to you? Explain

Part 10

In this scenario, there will be two times that the car and bike are in the same position, but they are so short due to difference in velocities that they will be too short to hand off the water.