Surface Area of a Cuboid: Find X When Surface Area = 250 cm²

To solve the problem, follow these steps:

• Step 1: Use the formula for the surface area of a cuboid:
  $S = 2(lw + lh + wh)$ where $l = x+3$, $w = 12-x$, $h = 5$.
• Step 2: Substitute the dimensions into the formula:
  $S = 2((x+3)(12-x) + (x+3) \cdot 5 + (12-x) \cdot 5)$.
• Step 3: Calculate each term:
  - $(x+3)(12-x) = x(12-x) + 3(12-x) = 12x - x^2 + 36 - 3x = -x^2 + 9x + 36$.
  - $(x+3) \cdot 5 = 5x + 15$.
  - $(12-x) \cdot 5 = 60 - 5x$.
• Step 4: Plug in these results into the surface area formula:
  $S = 2((-x^2 + 9x + 36) + (5x + 15) + (60 - 5x))$.
• Step 5: Simplify inside the parentheses:
  $-x^2 + 9x + 36 + 5x + 15 + 60 - 5x = -x^2 + 9x + 36 + 15 + 60 = -x^2 + 9x + 111$.
• Step 6: Complete the simplification:
  $S = 2(-x^2 + 9x + 111)$.
• Step 7: Solve for when $S = 250$ cm²:
  $250 = 2(-x^2 + 9x + 111)$.
• Step 8: Simplify and rearrange:
  $250 = -2x^2 + 18x + 222$, then
  $2x^2 - 18x + 28 = 0$ (after isolating zero on one side and simplifying).
• Step 9: Factor or use the quadratic formula to solve this equation:
  The equation factors to $2(x^2 - 9x + 14) = 0$,
  Then factor $x^2 - 9x + 14 = (x-7)(x-2) = 0$.
• Step 10: Solve for $x$:
  $x = 7$ or $x = 2$.
• Step 11: Verify that both values provide valid dimensions and confirm surface area.

The values of $x$ that satisfy the condition are $x = 2$ and $x = 7$.