Answer
300 cartons of food and 200 cartons of clothing.
Work Step by Step
Step 1. Assume $x$ cartons of food and $y$ cartons of clothing are shipped.
Step 2. Based on the given conditions, the number of people helped can be written as $z(x,y)=12x+5y$
Step 3. We can convert the constraints into inequalities as
$\begin{cases} x\geq0, y\geq0\\50x+20y\leq19,000\\20x+10y\leq8000 \end{cases}$
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
$z(0,0)=12(0)+5(0)=0$, $z(0,800)=12(0)+5(800)=4000$, $z(300,200)=12(300)+5(200)=4600$, $z(380,0)=12(380)+5(0)=4560$
Step 6. Based on the above results, we can find the maximum number of people helped as $z(300,200)= 4600$ with 300 cartons of food and 200 cartons of clothing.
