Answer
4 computation problems and 8 one-word problems will give a maximum score of 104.
Work Step by Step
Given that computation problem of 6 points is solved in 2 minutes and word problem of 10 points is solved in 4 minutes.
If total x number of computation problems and y number of the word problems are solved to achieve the maximum score.
Then, the possible objective function to get maximum score can be defined as:
\[z=6x+10y\].
Constraint of maximum time to take the test is 40 minutes, where x is solved in 2 minutes and y is solved in 4 minutes, which is given by the inequality:
\[\begin{align}
& 2x+4y\le 40 \\
& x+2y\le 20
\end{align}\]
Constraint that maximum number of problems can be solved are 12 can be defined by the inequality:
\[x+y\le 12\]
Thus, this is obvious from above two that\[x\ge 0,y\ge 0\].
Now try to maximize the value of objective function\[z=6x+10y\] according to the above constraints.
For this, find out the corner points from the linear equations of the above inequalities viz.
\[x+2y=20,x+y=12,x\ge 0,y\ge 0\]
Set 8 for y in the equations\[x+2y=20\]to get:
\[\begin{align}
& x+2.8=20 \\
& x+16=20 \\
& x=4
\end{align}\]
This provides vertex point as\[\left( 4,8 \right)\].
Again, set 0 for yin the equations \[x+y=12\]to get:
\[\begin{align}
& x+0=12 \\
& x=12
\end{align}\]
This provides x-intercepts as 12 and got the vertex points as\[\left( 12,0 \right)\].
Again, set 0 for x in the equations\[x+2y=20\]to get:
\[\begin{align}
& 0+2y=20 \\
& y=10
\end{align}\]
Thus, the x-intercepts is 10 and get the next vertex point as\[\left( 0,10 \right)\].
Now, 0 for x in the equation\[x+y=12\]to get:
\[\begin{align}
& 0+y=12 \\
& y=12
\end{align}\]
So, x-intercept obtained is 12 and the next vertex point is \[\left( 0,12 \right)\].
Vertex point \[\left( 0,0 \right)\]is the result of the intersection point of axis lines\[x=0,y=0\].
One more vertex point is possible from the intersection points of equations\[x+2y=20,x+y=12\]. So, solve these equations together.
\[\begin{align}
& x+y=12 \\
& x+2y=20
\end{align}\]
Subtract both the equations to obtain:
\[\begin{align}
& x+y-x-2y=12-80 \\
& -y=-8 \\
& y=8
\end{align}\]
Now substitute back 8 for y in any equation and solve for x:
\[\begin{align}
& x+8=12 \\
& x=12-8 \\
& =4
\end{align}\]
Thus, one more vertex point is\[\left( 4,8 \right)\].
Consider a test point\[\left( 0,0 \right)\]. This will be used to determine the shaded region.
For \[x+y\le 12\],
\[\begin{align}
& \left( 0 \right)+\left( 0 \right)\le 12 \\
& 0\le 12
\end{align}\]
As this inequality holds true, the shaded region would be towards the origin.
For \[x+2y\le 20\],
\[\begin{align}
& \left( 0 \right)+2\left( 0 \right)\le 20 \\
& 0\le 20
\end{align}\]
As this inequality holds true, the shaded region would be towards the origin.
