Answer
a. $a=-2, b=32, c=42$
b. 2pm,$170$ (ppm).
Work Step by Step
a. Step 1. Let $c=u, b=v, a=w$. The function becomes $y=u+vx+wx^2$. Use the data from the table to set up the system of equations:
$\begin{cases} u+2v+4w=98 \\ u+4v+16w=138 \\ u+10v+100w=162 \end{cases}$
Step 2. Write the matrix and perform row operations:
$\begin{bmatrix} 1 & 2 & 4 & | & 98 \\ 1 & 4 & 16 & | & 138 \\ 1 & 10 & 100 & | & 162 \end{bmatrix}\begin{array} .. \\R2-R1\to R2\\ R3-R1\to R3 \end{array}$
$\begin{bmatrix} 1 & 2 & 4 & | & 98 \\ 0 & 2 & 12 & | & 40 \\ 0 & 8 & 96 & | & 64 \end{bmatrix}\begin{array} .. \\R2/2 \to R2\\ R3/8\to R3 \end{array}$
$\begin{bmatrix} 1 & 2 & 4 & | & 98 \\ 0 & 1 & 6 & | & 20 \\ 0 & 1 & 12 & | & 8 \end{bmatrix}\begin{array} .. \\..\\ R3-R2\to R3 \end{array}$
$\begin{bmatrix} 1 & 2 & 4 & | & 98 \\ 0 & 1 & 6 & | & 20 \\ 0 & 0 & 6 & | & -12 \end{bmatrix}\begin{array} .. \\..\\ .. \end{array}$
Step 3. The last row gives $w=-2$; back-substitute to get $v=20-6(-2)=32$ and $u+2(32)+4(-2)=98, u=42$. Thus, we have $a=-2, b=32, c=42$ and the equation is $y=-2x^2+32x+42$
b. The maximum of the function can be found at $x=-\frac{b}{2a}=-\frac{32}{2(-2)}=8$ (hours after 6am, that is 2pm), which gives $y=-2(8)^2+32(8)+42=170$ (level of pollution in ppm).
