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Math Help - A cubic problem

  1. #1
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    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?

    Thanks for any help you can offer.
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  2. #2
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    Quote Originally Posted by dominicc View Post
    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 \\<br />
f'(1)=0 \\<br />
f(2)=2000  \\<br />
f(3)=0 \end{array}

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