Mathematics

Question

Please help me and show steps! I would really appreciate it!
Please help me and show steps! I would really appreciate it!

1 Answer

  • Answer:

    The aircraft has a height of 1000 m at t=2 sec, and at t=8 sec

    Step-by-step explanation:

    Finding Exact Roots Of Polynomials

    A polynomial can be expressed in the general form    

    [tex]\displaystyle p(x)=a_nx^n+a_{n-1}\ x^{n-1}+...+a_1\ x+a_0}[/tex]  

    The roots of the polynomial are the values of x for which    

    [tex]P(x)=0[/tex]

    Finding the roots is not an easy task and trying to find a general solution has been discussed for centuries. One of the best possible approaches is trying to factor the polynomial. It requires a good eye and experience, but it gives excellent results.    

    The function for the trajectory of an aircraft is given by    

    [tex]\displaystyle h(x)=0.5(-t^4+10t^3-216t^2+2000t-1200)[/tex]

    We need to find the values of t that make H=1000, that is

    [tex]\displaystyle 0.5(-t^4+10t^3-216t^2+2000t-1200)=1000[/tex]

    Dividing by -0.5

    [tex]\displaystyle t^4-10t^3+216t^2-2000t+1200=-2000[/tex]

    Rearranging, we set up the equation to solve

    [tex]\displaystyle t^4-10t^3+216t^2-2000t+3200=0[/tex]

    Expanding some terms

    [tex]\displaystyle t^4-8t^3-2t^3+200t^2+16t^2-1600t-400t+3200=0[/tex]

    Rearranging

    [tex]\displaystyle t^4-8t^3+200t^2-1600t-2t^3+16t^2-400t+3200=0[/tex]

    Factoring

    [tex]\displaystyle t(t^3-8t^2+200t-1600)-2(t^3-8t^2+200t-1600)=0[/tex]

    [tex]\displaystyle (t-2)(t^3-8t^2+200t-1600)=0[/tex]

    This produces our first root t=2. Now let's factor the remaining polynomial

    [tex]\displaystyle t^2(t-8)+200(t-8)=0[/tex]

    [tex]\displaystyle (t^2+200)(t-8)=0[/tex]

    This gives us the second real root t=8. The other two roots are not real numbers, so we only keep two solutions

    [tex]\displaystyle t=2,\ t=8[/tex]

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