Surface Area of a Cuboid: Extended distributive law

To find $X$, follow these steps:

• Step 1: Apply the formula for the surface area of a cuboid: $SA = 2(lw + lh + wh)$.
• Step 2: Substitute the given dimensions: $(X+4)$, $7$, and $8$.
• Step 3: Set up the equation given $SA = 292$.

Using the surface area formula:

$SA = 2\left((X+4) \times 7 + (X+4) \times 8 + 7 \times 8\right) = 292$

Calculate each part:

$(X+4) \times 7 = 7X + 28$

$(X+4) \times 8 = 8X + 32$

$7 \times 8 = 56$

Combine and simplify:

$7X + 28 + 8X + 32 + 56 = 15X + 116$

Use the equation for surface area:

$2(15X + 116) = 292$

$30X + 232 = 292$

Simplifying further:

$30X = 292 - 232$

$30X = 60$

$X = \frac{60}{30}$

$X = 2$

Therefore, the value of $X$ is $2$.

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 $x = 7$.

To solve this problem, we'll calculate the surface area of the cuboid and solve for the unknown dimension $X$ using the formula for the surface area of a cuboid.

Given dimensions are:

• Length: $20 - X$
• Width: $11 \, \text{cm}$
• Height: $13 \, \text{cm}$

Using the formula for the surface area of a cuboid: $SA = 2(lw + lh + wh)$ Substitute the values: $862 = 2((20-X) \times 11 + (20-X) \times 13 + 11 \times 13)$

First, simplify the more straightforward terms: $11 \times 13 = 143$ Next, distribute and simplify: $(20-X) \times 11 = 220 - 11X$ $(20-X) \times 13 = 260 - 13X$ $SA = 2((220 - 11X) + (260 - 13X) + 143)$ $SA = 2(623 - 24X)$

Set this equation equal to the surface area: $862 = 2(623 - 24X)$ $862 = 1246 - 48X$ Rearranging gives: $48X = 1246 - 862$ $48X = 384$ Solving for $X$: $X = \frac{384}{48} = 8$

Therefore, the value of $X$ is 8 cm.

To solve for $X$, we begin with the formula for the surface area of a cuboid:

$A = 2(ab + bc + ca)$, where $a = 10$, $b = (X - 3)$, and $c = 12$.

Substitute the given surface area into the formula:

$284 = 2((10)(X-3) + (X-3)(12) + (10)(12))$

Simplify the equation by first expanding each term inside the parentheses:

$284 = 2(10X - 30 + 12X - 36 + 120)$

Combine like terms:

$284 = 2(22X + 54)$

Divide both sides by 2 to isolate the linear expression:

$142 = 22X + 54$

Subtract 54 from both sides to solve for $22X$:

$88 = 22X$

Finally, divide by 22 to find $X$:

$X = \frac{88}{22} = 4$

Therefore, the solution to the problem is $X = 4$.