Calculate X in a Cuboid with Surface Area 102: Given Dimensions 7 and 3

To solve this problem, we will use the formula for the surface area of a cuboid:

$S = 2(lw + lh + wh)$

Given:

• Surface area, $S = 102$
• Length, $l = 7$
• Width, $w = 3$
• Height, $h = X$

Substitute the known values into the surface area formula:

$102 = 2(7 \cdot 3 + 7 \cdot X + 3 \cdot X)$

Simplify by calculating the known products:

$102 = 2(21 + 7X + 3X)$

Combine like terms:

$102 = 2(21 + 10X)$

Distribute the 2 across the terms inside the parentheses:

$102 = 42 + 20X$

To isolate $X$, subtract 42 from both sides:

$60 = 20X$

Finally, divide both sides by 20 to solve for $X$:

$X = \frac{60}{20} = 3$

Therefore, the solution to the problem is $X = 3$.