Functions of Several Variables — Question 10

Question 10 – Solution

Strategy Interpret the three positive factors $x$, $y$, and $h=12-x-y$, then apply AM–GM.

Step 1: Domain Positivity requires $$\boxed{x>0,\qquad y>0,\qquad x+y<12}.$$ This is the interior of a triangular region.

Step 2: Bound The three positive numbers $x,y,h$ have fixed sum $12$. AM–GM gives $$\frac{x+y+h}{3}\ge\sqrt[3]{xyh}\Longrightarrow 4\ge\sqrt[3]{V}.$$ Cubing yields $V\le 64$.

Step 3: Equality Equality in AM–GM occurs exactly when $$x=y=h=4.$$ Thus the unique maximizing dimensions are $\boxed{x=4,y=4,h=4}$ and $$\boxed{V_{\max}=64}.$$ Direct substitution satisfies the material constraint and confirms the volume.