Answer
4 computation problems and 8 word problems; maximum $z(4,8)= 104$
Work Step by Step
Step 1. Assume $x$ number of computation problems and $y$ number of word problems are needed.
Step 2. Based on the given conditions, the total score can be written as $z(x,y)=6x+ 10y$
Step 3. We can convert the constraints into inequalities as $\begin{cases} x\geq0,y\geq0\\ 2x+4x\leq40\\x+y\leq12 \end{cases}$
Step 4. Graphing the above inequalities, we can find the vertices and the solution region as a four-sided area in the first quadrant.
Step 5. With the objective equation and vertices, we have
$z(0,0)=6(0)+10(0)=0$, $z(0,10)=6(0)+10(10)=100$, $z(4,8)=6(4)+10(8)=104$, $z(12,0)=6(12)+10(0)=72$,
Step 6. Based on the above results, we can find the maximum as $z(4,8)= 104$ with 4 computation problems and 8 word problems.
