To solve this problem, let’s go through it step-by-step.

Given:

• Material available for making a box with a square base and open top: $1875 \, \text{cm}^2$
• We need to find the largest possible volume of this box.

Let:

• $x$ be the length of a side of the square base.
• $h$ be the height of the box.

Step 1: Surface Area Constraint

Since the box has an open top, the surface area used is the area of the base plus the area of the four sides. This gives: $x^2 + 4xh = 1875$

Step 2: Volume of the Box

The volume $V$ of the box is given by: $V = x^2 \cdot h$

Step 3: Express $h$ in Terms of $x$

From the surface area equation: $4xh = 1875 - x^2$ $h = \frac{1875 - x^2}{4x}$

Step 4: Substitute $h$ in the Volume Formula

Substitute $h$ from the equation above into the volume formula: $V = x^2 \cdot \frac{1875 - x^2}{4x}$ $V = \frac{x(1875 - x^2)}{4}$ $V = \frac{1875x - x^3}{4}$

Step 5: Maximize the Volume Function

Now, differentiate $V$ with respect to $x$ and set the derivative to zero to find the critical points.

$\frac{dV}{dx} = \frac{1875 - 3x^2}{4}$

Setting $\frac{dV}{dx} = 0$: $1875 - 3x^2 = 0$ $3x^2 = 1875$ $x^2 = 625$ $x = 25$

Step 6: Find $h$ when $x = 25$

Substitute $x = 25$ back into the equation for $h$: $h = \frac{1875 - 25^2}{4 \times 25}$ $h = \frac{1875 - 625}{100}$ $h = \frac{1250}{100} = 12.5$

Step 7: Calculate the Maximum Volume

Now, substitute $x = 25$ and $h = 12.5$ into the volume formula: $V = x^2 \cdot h = 25^2 \cdot 12.5$ $V = 625 \cdot 12.5 = 7812.5 \, \text{cm}^3$

Answer:

The largest possible volume of the box is $7812.5 \, \text{cm}^3$.

Would you like further details on any specific step, or have any additional questions?

Here are some related questions:

1. How does changing the amount of material affect the maximum volume?
2. What if the box had a closed top—how would that change the calculations?
3. How would the dimensions change if the surface area constraint were different?
4. Could we use a different method, such as the Lagrange multiplier, to solve this problem?
5. What happens to the volume if the height is increased while keeping the base area the same?

Tip: Always start with identifying constraints when maximizing or minimizing a function in problems involving geometry and optimization.