The diagram shows a cat scratch stand which consists of a solid cylinder fixed to a solid cuboid. The cylinder has a radius of $x\text{ cm}$ and a height of $y\text{ cm}$. The cuboid has a square base of side $3x\text{ cm}$ and a height of $x\text{ cm}$. Given that $x$ can vary,

1. Formulate the expression for the total volume of the stand

$\text{Volume of cylinder} = \pi x^2 y$
$\text{Volume of cuboid} = (3x)^2(x) = 9x^3$
$\text{Total volume} = \pi x^2 y + 9x^3$

2. Equate the total volume to 1300 and make $y$ the subject

$\pi x^2 y + 9x^3 = 1300$
$\pi x^2 y = 1300 - 9x^3$
$y = \frac{1300 - 9x^3}{\pi x^2}$

1. Recall the expression for $y$ from part (i)

From part (i), the volume is $V = \pi x^2 y + (3x)(3x)(x) = 1300$.
$\pi x^2 y + 9x^3 = 1300$
$y = \frac{1300 - 9x^3}{\pi x^2}$

2. Formulate the expression for the total surface area of the stand

The total surface area $A$ consists of:

• The curved surface area of the cylinder: $2\pi x y$
• The top circular face of the cylinder: $\pi x^2$
• The exposed top face of the cuboid: $(3x)^2 - \pi x^2 = 9x^2 - \pi x^2$
• The four rectangular side faces of the cuboid: $4 \times (3x \times x) = 12x^2$
• The bottom square face of the cuboid: $(3x)^2 = 9x^2$

Total area $A = 2\pi x y + \pi x^2 + (9x^2 - \pi x^2) + 12x^2 + 9x^2$
$A = 2\pi x y + 30x^2$

3. Substitute $y$ into the area expression to show the given formula

Substitute $y = \frac{1300 - 9x^3}{\pi x^2}$ into $A$:
$A = 2\pi x \left( \frac{1300 - 9x^3}{\pi x^2} \right) + 30x^2$
$A = \frac{2(1300 - 9x^3)}{x} + 30x^2$
$A = \frac{2600}{x} - \frac{18x^3}{x} + 30x^2$
$A = \frac{2600}{x} - 18x^2 + 30x^2$
$A = \frac{2600}{x} + 12x^2$ (shown)

1. Find the second derivative of $A$ with respect to $x$

From part (iii), $\frac{dA}{dx} = -2600x^{-2} + 24x$.

$\frac{d^2A}{dx^2} = \frac{d}{dx}(-2600x^{-2} + 24x)$
$\frac{d^2A}{dx^2} = 5200x^{-3} + 24$

2. Evaluate the sign of the second derivative to determine the nature of the stationary value

Since $x$ is a length, $x > 0$.
Therefore, $\frac{d^2A}{dx^2} = \frac{5200}{x^3} + 24 > 0$.
Since the second derivative is positive, the stationary value is a minimum.