Question

1. the table below shows the height of a model rocket launched during a science fair. create a quadratic regression model equation to find the height after 2.8 seconds.

time (seconds)	1	3	5	6
height (feet)	160	192	96	0

Explanation:

Step1: Recall quadratic regression formula

A quadratic regression model has the form $y = ax^{2}+bx + c$. We can use a system of equations based on the data points $(x_i,y_i)$ from the table. For the points $(1,160)$, $(3,192)$, $(5,96)$ and $(6,0)$:
When $x = 1,y=160$, we have $a\times1^{2}+b\times1 + c=160$, so $a + b + c=160$.
When $x = 3,y = 192$, we have $a\times3^{2}+b\times3 + c=192$, so $9a+3b + c=192$.
When $x = 5,y = 96$, we have $a\times5^{2}+b\times5 + c=96$, so $25a+5b + c=96$.

Step2: Subtract equations to eliminate $c$

Subtract the first - equation from the second:
$(9a + 3b + c)-(a + b + c)=192 - 160$
$9a+3b + c - a - b - c=32$
$8a+2b=32$, simplify to $4a + b=16$ (Equation 1).
Subtract the second equation from the third:
$(25a+5b + c)-(9a + 3b + c)=96 - 192$
$25a+5b + c - 9a - 3b - c=-96$
$16a+2b=-96$, simplify to $8a + b=-48$ (Equation 2).

Step3: Solve for $a$ and $b$

Subtract Equation 1 from Equation 2:
$(8a + b)-(4a + b)=-48 - 16$
$8a + b-4a - b=-64$
$4a=-64$, so $a=-16$.
Substitute $a = - 16$ into $4a + b=16$:
$4\times(-16)+b=16$
$-64 + b=16$, so $b = 80$.

Step4: Solve for $c$

Substitute $a=-16$ and $b = 80$ into $a + b + c=160$:
$-16+80 + c=160$
$64 + c=160$, so $c = 96$.
The quadratic regression equation is $y=-16x^{2}+80x + 96$.

Step5: Find the height at $x = 2.8$

Substitute $x = 2.8$ into $y=-16x^{2}+80x + 96$:
$y=-16\times(2.8)^{2}+80\times2.8 + 96$
$y=-16\times7.84+224 + 96$
$y=-125.44+224 + 96$
$y=194.56$.

Answer:

$194.56$ feet