A tinsmith in Nairobi has a rectangular sheet of metal, 24 cm by 15 cm…

Step 1. The volume. $V = x(24 - 2x)(15 - 2x) = 4x^3 - 78x^2 + 360x$, for $0 < x < 7.5$.

Step 2. The derivative. $V' = 12x^2 - 156x + 360 = 12(x - 3)(x - 10)$.

Step 3. The critical points. $x = 3$ or $x = 10$. The value $10$ is outside the domain, because the width $15 - 2x$ would be negative.

Step 4. The check. $V'' = 24x - 156$, which is $-84 < 0$ at $x = 3$. So it is a maximum.

Step 5. The volume. $V(3) = 3 \times 18 \times 9 = 486\ \text{cm}^3$.