1. ## A cubic problem

A ‘rogue satellite’ has its distance from Earth, d thousand kilometres, modelled by a cubic function of time, t days after launch. After 1 day it reaches a maximum distance from Earth of 4000 kilometres, then after 2 days it is 2000 kilometres away. It effectively returns to Earth after 3 days, then moves further and further away.

a What is the satellite’s initial distance from Earth?
b Sketch the graph of d versus t for the first 6 days of travel.
c Express d as a function of t.

Hello, I'm a 17 year old student from australia and was wondering if anyone can help me with this problem. A and B were easy, but i have no idea how to do C. How do you create a cubic function with that information given? Is there a certain fomular or something?

2. Originally Posted by dominicc
A ‘rogue satellite’ has its distance from Earth, d thousand kilometres, modelled by a cubic function of time, t days after launch. After 1 day it reaches a maximum distance from Earth of 4000 kilometres, then after 2 days it is 2000 kilometres away. It effectively returns to Earth after 3 days, then moves further and further away.

a What is the satellite’s initial distance from Earth?
b Sketch the graph of d versus t for the first 6 days of travel.
c Express d as a function of t.

...
1. A cubic function in t with the coefficients a, b, c, d is:

$f(t)=at^3+bt^2+ct+d$ . It has the first derivation:

$f'(t)=3at^2+2bt+c$

2. From the text of the question you know:

$\begin{array}{l}f(1)=4000 \\
f'(1)=0 \\
f(2)=2000 \\
f(3)=0 \end{array}$

3. Obviously you have to solve a system of simultaneous equations. Solve for (a, b, c, d).