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 2 - 2007 - Paper 2

To find the maximum volume, we will first differentiate $V$ with respect to $x$:

$\frac{dV}{dx} = 200\left(\frac{-4x^2(600 - 4x^2)^{\prime}}{3}\right).$

Set the derivative equal to zero to find the critical points:

$0 = 200 - 8x^2 \implies x^2 = 25 \implies x = 5.$

Substituting $x = 5$ back into the equation for $y$ gives:

$y = \frac{600 - 4(5^2)}{6(5)} = \frac{600 - 100}{30} = \frac{500}{30} \approx 16.67.$

Now substituting these values into the volume:

$V = \frac{200 \cdot 5 \cdot 16.67^2}{3} = 943.34 \approx 943 \text{ cm}^3$

To confirm that we've found a maximum, we calculate the second derivative:

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

Substituting our critical point $x = 5$:

$\frac{d^2V}{dx^2} = -8(5) = -40.$

Since this value is negative, it indicates that the function is concave down, confirming that we have a local maximum at this point.