Answer
20 American airplanes and 24 British airplanes.
Work Step by Step
Step 1. Assume $x$ number of American airplanes and $y$ number of British airplanes.
Step 2. Based on the given conditions, the total cargo capacity can be written as $z(x,y)=30000x+20000y$
Step 3. We can convert the constraints into inequalities as
$\begin{cases} x\geq0,y\geq0\\x+y\leq44\\16x+8y\leq512\\9000x+5000y\leq300000 \end{cases}$
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
$z(0,0)=30000(0)+20000(0)=0$, $z(0,44)=30000(0)+20000(44)=880,000$, $z(20,24)=30000(20)+20000(24)=1,080,000$, $z(32,0)=30000(32)+20000(0)=960,000$,
Step 6. Based on the above results, we can find the maximum cargo capacity as
$z(20,24)=30000(20)+20000(24)=1,080,000$
with 20 American airplanes and 24 British airplanes.
