Question

From a square of side 30 cm the squares of side x cm is cut off to make a cuboid of maximum volume. The surface area of cuboid with open top is?

• A. 400 cm²
• B. 464 cm²
• C. 800 cm²
• D. 900 cm²

Answer 1

The Correct Option is C

Given:

Let \( x \) be the side length of the square cut from each corner of the square sheet.

• Step 1: Dimensions of the open box:
  • Length: \( 30 - 2x \)
  • Breadth: \( 30 - 2x \)
  • Height: \( x \)
  The volume \( V(x) \) of the box is:

  \( V(x) = (30 - 2x)^2x = 4x^3 - 120x^2 + 900x \).

• Step 2: Maximize the volume by finding the critical points:

  \( V'(x) = \frac{dV}{dx} = 12x^2 - 240x + 900 \).

  Set \( V'(x) = 0 \):

  \( 0 = 12x^2 - 240x + 900 \).

  Simplify:

  \( 0 = x^2 - 20x + 75 \).

  Factorize:

  \( 0 = (x - 5)(x - 15) \).

  So, \( x = 5 \) or \( x = 15 \).
• Step 3: Eliminate non-feasible solutions:

  If \( x = 15 \), the length and breadth become \( 30 - 2(15) = 0 \), which is not possible. Therefore, \( x = 5 \).

• Step 4: Surface area of the open box (without the top):

  \( S(x) = (30 - 2x)^2 + 4x(30 - 2x) \).

  Substituting \( x = 5 \):

  \( S(5) = (30 - 2(5))^2 + 4(5)(30 - 2(5)) \).

  Simplify:

  \( S(5) = (20)^2 + 20(20) = 400 + 400 = 800 \, \text{cm}^2 \).

Final Answer: The surface area of the open box is \( 800 \, \text{cm}^2 \).