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]
