Step-by-Step Solution

Topic: Maxima and Minima (Optimization)

(I) EXPRESSION FOR VOLUME V

When squares of side \( x \) are cut from corners, the dimensions become:

  • Length \( L = 36 – 2x \)
  • Breadth \( B = 36 – 2x \)
  • Height \( H = x \)

Volume \( V = L \times B \times H \):

\( V = (36 – 2x)^2 \cdot x \)

Expanding this for easier differentiation:

\( V = x(1296 – 144x + 4x^2) \)
\( V = 4x^3 – 144x^2 + 1296x \)

(II) FINDING THE DERIVATIVE dV/dx

Differentiating \( V = 4x^3 – 144x^2 + 1296x \) with respect to \( x \):

\[ \frac{dV}{dx} = 12x^2 – 288x + 1296 \]

Simplified by factoring out 12:

\( \frac{dV}{dx} = 12(x^2 – 24x + 108) \)

(III) MAXIMIZING THE VOLUME

For maximum volume, set \( \frac{dV}{dx} = 0 \):

\( 12(x^2 – 24x + 108) = 0 \)
\( x^2 – 24x + 108 = 0 \)

Solving the quadratic equation \( (x – 6)(x – 18) = 0 \):

\( x = 6 \) or \( x = 18 \)

Validation: If \( x = 18 \), the length \( 36 – 2(18) = 0 \), which is impossible. So, we reject \( x = 18 \). Thus, \( x = 6 \) cm.

(Optional) Maximum Volume Calculation:

\( V(6) = (36 – 2(6))^2 \cdot 6 \)
\( V(6) = (24)^2 \cdot 6 = 576 \cdot 6 = \mathbf{3456 \text{ cm}^3} \)

⚠️ Silly Mistake Audit: The “Impossible” Cut

Always check the domain of your variable! You mathematically found two values: \( x=6 \) and \( x=18 \). However, if you cut 18 cm from both corners of a 36 cm sheet, you are left with \( 36 – 18 – 18 = 0 \) cm of board. The box disappears! Therefore, \( x=18 \) must be rejected.

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