Figure 4 shows a solid brick in the shape of a cuboid measuring $2x$ cm by $x$ cm by $y$ cm - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 2

To find the maximum value of $V$, we first need to differentiate $V$ with respect to $x$:

$\frac{dV}{dx} = 200 - 4x^2.$

We set the derivative to zero to find critical points:

$200 - 4x^2 = 0\Rightarrow 4x^2=200\Rightarrow x^2=50\Rightarrow x=\sqrt{50} = 5\sqrt{2}.$

Next, we calculate the volume at this critical point:

$V = 200(5\sqrt{2}) - \frac{4}{3}(5\sqrt{2})^3$

Calculating gives:

$V = 200(5\sqrt{2}) - \frac{4}{3}(125\cdot 2 \sqrt{2}) = 1000\sqrt{2} - \frac{1000\sqrt{2}}{3} = \frac{3000\sqrt{2}}{3} - \frac{1000\sqrt{2}}{3} = \frac{2000\sqrt{2}}{3}.$

Now let's evaluate $\frac{2000\sqrt{2}}{3}$:

Calculating gives approximately $943$ cm$^3$.

To confirm that the value found is a maximum, we will check the second derivative:

$\frac{d^2V}{dx^2} = -8x.$

At the critical point $x = 5\sqrt{2}$:

$\frac{d^2V}{dx^2} = -8(5\sqrt{2}) < 0,$

indicating that the function is concave down at this point. Thus, $V$ has a local maximum at this critical point, justifying that the volume we found is indeed a maximum.