Answer
$y=\dfrac{5}{8}x^2-50x+1150$
Work Step by Step
Consider the function
$y=ax^2+bx+c$
Build a system of equations to determine $a,b,c$:
$\begin{cases}
a(20^2)+b(20)+c=400\\
a(40^2)+b(40)+c=400\\
a(60^2)+b(60)+c=400
\end{cases}$
$\begin{cases}
400a+20b+c=400\\
1600a+40b+c=150\\
3600a+60b+c=400
\end{cases}$
In order to use Cramer's Rule, first we have to compute the determinants $D,D_a,D_b,D_c$:
$D=\begin{vmatrix}400&20&1\\1600&40&1\\3600&60&1\end{vmatrix}=16000+72000+96000-144000-24000-32000=-16000$
$D_a=\begin{vmatrix}400&20&1\\150&40&1\\400&60&1\end{vmatrix}=16000+8000+9000-16000-24000-3000=-10000$
$D_b=\begin{vmatrix}400&400&1\\1600&150&1\\3600&400&1\end{vmatrix}=60000+1440000+640000-540000-160000-640000=800000$
$D_c=\begin{vmatrix}400&20&400\\1600&40&150\\3600&60&400\end{vmatrix}=6400000+10800000+38400000-57600000-360000-12800000=-18400000$
We use Cramer's Rule to determine the solutions of the system:
$a=\dfrac{D_a}{D}=\dfrac{-10000}{-160000}=\dfrac{5}{8}$
$b=\dfrac{D_b}{D}=\dfrac{800000}{-16000}=-50$
$c=\dfrac{D_c}{D}=\dfrac{-18400000}{-16000}=1150$
The solution is:
$a=\dfrac{5}{8}$
$b=-50$
$c=1150$
The function is:
$y=\dfrac{5}{8}x^2-50x+1150$
Compute $y$ for $x=30$:
$y=\dfrac{5}{8}(30^2)-50(30)+1150=212.5$
Compute $y$ for $x=50$:
$y=\dfrac{5}{8}(50^2)-50(50)+1150=212.5$
