To solve this problem, we need to maximize the volume of the box given the constraints on surface area.

Step-by-Step Solution

1. Define Variables:

   • Let $x$ be the length of each side of the square base (in cm).
   • Let $y$ be the height of the box (in cm).

2. Volume of the Box:

   • The volume $V$ of the box can be written as: $V = x^2 \cdot y$

3. Surface Area Constraint:

   • The box has a square base and an open top.
   • The total surface area includes the area of the base ($x^2$) and the four sides of the box ($4 \cdot x \cdot y$).
   • Given that the total surface area is 2300 cm², we have: $x^2 + 4 \cdot x \cdot y = 2300$

4. Express $y$ in Terms of $x$:

   • Rearrange the surface area equation to solve for $y$: $4 \cdot x \cdot y = 2300 - x^2$ $y = \frac{2300 - x^2}{4x}$

5. Substitute $y$ in the Volume Equation:

   • Substitute $y$ from the previous step into the volume equation: $V = x^2 \cdot \frac{2300 - x^2}{4x}$ Simplify to: $V = \frac{2300x - x^3}{4}$

6. Differentiate $V$ with Respect to $x$ and Set to Zero:

   • To maximize the volume, take the derivative of $V$ with respect to $x$ and set it equal to zero: $\frac{dV}{dx} = \frac{2300 - 3x^2}{4} = 0$
   • Solve for $x$: $2300 = 3x^2$ $x^2 = \frac{2300}{3}$ $x = \sqrt{\frac{2300}{3}} \approx 27.75 \, \text{cm}$

7. Calculate $y$ Using $x$:

   • Substitute $x \approx 27.75$ back into the equation for $y$: $y = \frac{2300 - (27.75)^2}{4 \times 27.75} \approx 13.88 \, \text{cm}$

8. Calculate the Maximum Volume:

   • Now, substitute $x \approx 27.75$ and $y \approx 13.88$ into the volume formula: $V = 27.75^2 \cdot 13.88 \approx 10690.2 \, \text{cubic centimeters}$

Answer

The largest possible volume of the box is approximately: $\boxed{10690.2 \, \text{cm}^3}$

Would you like a breakdown of each step or further clarification?

Related Questions

1. How do you find the critical points of a function to maximize or minimize values?
2. How can you set up constraints in optimization problems with different shapes?
3. What are the practical applications of maximizing volume in real-world design?
4. How does calculus help in solving real-world problems with constraints?
5. What other methods, besides differentiation, can help solve optimization problems?

Tip

For optimization problems, always ensure the constraint equation is correctly set up and simplify it before differentiating.