A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$$^2$$) is equal to

A square piece of tin of side 30 cm. Cut squares of side $$x$$ from each corner.

After folding: length = width = $$30 - 2x$$, height = $$x$$.

Volume: $$V = x(30-2x)^2$$

$$\frac{dV}{dx} = (30-2x)^2 + x \cdot 2(30-2x)(-2) = (30-2x)[(30-2x) - 4x] = (30-2x)(30-6x)$$

Setting $$\frac{dV}{dx} = 0$$: $$x = 15$$ (rejected, gives V=0) or $$x = 5$$.

At $$x = 5$$: length = width = 20 cm, height = 5 cm.

Surface area (without top) = base + 4 sides:

$$= 20 \times 20 + 4 \times (20 \times 5) = 400 + 400 = 800 \text{ cm}^2$$

The correct answer is Option 1: 800.