ANSWER:
19.
[tex]h(x)=-\frac{2}{3}\cdot x^3+12\cdot x^2-\frac{334}{3}\cdot x+500[/tex]20. 120 m
STEP-BY-STEP EXPLANATION:
Let h(x) denote height at any bounce x:
[tex]h(x)=ax^3+bx^2+cx+d[/tex]When x = 0
[tex]\begin{gathered} a\cdot0^3+b\cdot0^2+c\cdot0+d=500 \\ d=500 \end{gathered}[/tex]When x = 1
[tex]\begin{gathered} a\cdot1^3+b\cdot1^2+c\cdot1+d=400 \\ a+b+c+d=400 \end{gathered}[/tex]When x = 2
[tex]\begin{gathered} a\cdot2^3+b\cdot2^2+c\cdot2+d=320 \\ 8a+4b+2c+d=320 \end{gathered}[/tex]When x = 3
[tex]\begin{gathered} a\cdot3^3+b\cdot3^2+c\cdot3+d=256 \\ 27a+9b+3c+d=256 \end{gathered}[/tex]Now, we calculate 3 equations by subtracting the value of d, in order to calculate the value of a, b and c, like this:
[tex]\begin{gathered} a+b+c+d-d=400-500 \\ a+b+c=-100\rightarrow a=-100-b-c\text{ (1)} \\ \\ 8a+4b+2c+d-d=320-500 \\ 8a+4b+2c=-180\text{ (2)} \\ \\ 27a+9b+3c+d-d=256-500 \\ 27a+9b+3c=-244\text{ (3)} \end{gathered}[/tex]We solve the system of equations as follows:
Replacing (1) in (2) and (3):
[tex]\begin{gathered} 8\cdot(-100-b-c)+4b+2c=-180\rightarrow-4b-6c-800=-180\rightarrow b=-\frac{3c+310}{2}(4) \\ 27\cdot(-100-b-c)+9b+3c=-244\rightarrow-18b-24c-2700=-244\text{ (5)} \end{gathered}[/tex]Replacing (4) in (5) and solving for c:
[tex]\begin{gathered} -18\cdot\mleft(-\frac{3c+310}{2}\mright)-24c-2700=-244 \\ 27c+2790-24c-2700=-244 \\ 3c+90=-244 \\ c=-\frac{334}{3} \end{gathered}[/tex]We replace the value of c in (4):
[tex]b=-\frac{3\cdot-\frac{334}{3}+310}{2}=-\frac{-334+310}{2}=-\frac{24}{2}=-12[/tex]Now we replace the values of b and c in (1):
[tex]\begin{gathered} a=-100-\mleft(12\mright)-\mleft(-\frac{334}{3}\mright) \\ a=-\frac{2}{3} \end{gathered}[/tex]Therefore, the a rule to represent the height of then ball after each bounce is the following:
[tex]h(x)=-\frac{2}{3}\cdot x^3+12\cdot x^2-\frac{334}{3}\cdot x+500[/tex]The height after the sixth bounce , would be when x = 6, we replace:
[tex]\begin{gathered} h(6)=-\frac{2}{3}\cdot6^3+12\cdot6^2-\frac{334}{3}\cdot6+500 \\ h(6)=-\frac{2}{3}\cdot216+12\cdot36-334\cdot2+500 \\ h(6)=-144+432-668+500 \\ h(6)=120 \end{gathered}[/tex]The height after the sixth bounce is 120 meters
