10. You plan to make a simple, open topped box from a piece of sheet metal by cutting a square - of equal size from each corner and folding up the sides as shown in the diagram: If $l=200 \mathrm{~mm}$ and $w=150 \mathrm{~mm}$ calculate: a) The value of $x$ which will give the maximum volume b) The maximum volume of the box c) Comment of the value obtained in part b.

Step 1 ：Let the size of the square cut from each corner be x. The volume of the box can be represented as a function of x: \(V(x) = x * (l - 2x) * (w - 2x)\), where l is the length, w is the width, and x is the size of the square cut from each corner.

Step 2 ：Find the maximum volume by taking the derivative of the volume function with respect to x and setting it equal to zero: \(\frac{dV}{dx} = -2x(150 - 2x) - 2x(200 - 2x) + (150 - 2x)(200 - 2x)\).

Step 3 ：Solve for x to find the value that maximizes the volume: critical points are \(\frac{175}{3} - \frac{25\sqrt{13}}{3}\) and \(\frac{25\sqrt{13}}{3} + \frac{175}{3}\).

Step 4 ：Find the maximum volume: \(V_{max} = \left(\frac{100}{3} + \frac{50\sqrt{13}}{3}\right)\left(\frac{175}{3} - \frac{25\sqrt{13}}{3}\right)\left(\frac{50\sqrt{13}}{3} + \frac{250}{3}\right)\).

Step 5 ：\(\boxed{\text{a)}\ x \approx \frac{175}{3} - \frac{25\sqrt{13}}{3}}\)

Step 6 ：\(\boxed{\text{b)}\ V_{max} \approx \left(\frac{100}{3} + \frac{50\sqrt{13}}{3}\right)\left(\frac{175}{3} - \frac{25\sqrt{13}}{3}\right)\left(\frac{50\sqrt{13}}{3} + \frac{250}{3}\right)}\)

Step 7 ：\(\boxed{\text{c)}\ \text{The value obtained in part b represents the maximum volume that the box can hold when the square cut from each corner has the optimal size. This is useful for maximizing the efficiency of the box's design and ensuring that it can hold as much as possible given its dimensions.}}\)