Mathematics

When you cut squares from the corners and fold up, you're trading away corner material to create height—the key is that the base shrinks while the box gets taller. The volume function $V(x) = x(60-2x)(30-2x)$ captures this trade-off: $x$ is your height, and $(60-2x)$ and $(30-2x)$ are the resulting base dimensions. To maximize, find where $\frac{dV}{dx} = 0$, then verify it's a maximum by checking the second derivative or the domain constraint ($x < 15$).

When you cut squares from the corners and fold up, you're trading away corner material to create height—the key is that the base shrinks while the box gets taller. The volume function $V(x) = x(60-2x)(30-2x)$ captures this trade-off: $x$ is your height, and $(60-2x)$ and $(30-2x)$ are the resulting base dimensions. To maximize, find where $\frac{dV}{dx} = 0$, then verify it's a maximum by checking the second derivative or the domain constraint ($x < 15$).