Question

part A assume the ball passes through the points (3,8) , (5, 20/3) and (6,5). use this data to set up a system of three equations and three unknowns (a, b, and c) that will allow you to find the equation of a parabola. write the system in the space provided.part Buse Matrix manipulation to solve for a, b, and c. Set up a matrix equation for AX=B based on the system of equations you delivered in Part B where X is a matrix of the variables a, b, and c. then, use Gauss-Jordan elimination to find the inverse of A. finally, use your results to write the equation of the parabola show your work and final equation in the space provided.part Cnow that you have determined the equation of the parabola, assume that X represents the number of seconds that have passed since the ball was thrown and determine approximately how long it will take for the ball to hit the ground.

Answer 1

Given the three end points (3,8), (5,20/3) and (6,5) which are written in the form of (x1,y1), (x2,y2) and (x3,y3).

Write out the formula to obtain the equation of the parabola

`y=ax^2+bx+c`

Write out the given coordinates

`\begin{gathered} x_1=3,y_1=\text{ 8} \\ x_2=5,y_2=\text{ }(20)/(3) \\ x_3=6,y_3=\text{ 5} \end{gathered}`

Substitute the coordinates into the given equation above to obtain the three equations

`\begin{gathered} y_1=ax^2_1+bx_1+c \\ 8=a(3)^2+\text{ b(3)+ c} \\ 8=\text{ 9a +3b+c }\ldots\ldots\ldots\ldots equation\text{ 1} \end{gathered}`
`\begin{gathered} y_2=ax^2_{^{}2}+bx_2+c \\ (20)/(3)=a(5)^2_{}+\text{ b(5)+ c} \\ (20)/(3)=\text{ 25a+5b+c} \\ \text{Multiply all through by 3, we have} \\ 20=\text{ 3}*25a+3*5b+3* c \\ 20=\text{ 75a+15b+3c }\ldots\ldots\ldots\ldots\ldots equation\text{ 2} \end{gathered}`
`\begin{gathered} y_3=ax^2_3+bx_3+\text{ c} \\ 5_{}=a(6)^2+\text{ b(6)+c} \\ 5=\text{ 36a+ 6b+c }\ldots\ldots\ldots\ldots\ldots\ldots\ldots equation\text{ 3} \end{gathered}`

Combining the three equations together

`\begin{gathered} 9a+3b+c=8\ldots\ldots\ldots\text{.equation 1} \\ 75a\text{ +15b+3c=20}\ldots\ldots\ldots equation\text{ 2} \\ 36a+6b+c=5\ldots\ldots\ldots\text{.equation 3} \end{gathered}`

Hence, the three equations above are the equations of the parabola with three unknowns a,b and c.