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

Surface Area Formula with Unknown Dimensions

A cuboid has a surface area of 102.

Calculate X.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find X
00:03 Use the formula to calculate the surface area of a box
00:10 2 times (sum of face areas)
00:20 Substitute appropriate values and solve for X
00:48 Solve each multiplication separately and sum
01:08 Open parentheses properly, multiply by each factor
01:18 Isolate X
01:28 And that's the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

A cuboid has a surface area of 102.

Calculate X.

XXX777333

2

Step-by-step solution

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

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

Given:

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

Substitute the known values into the surface area formula:

102=2(73+7X+3X) 102 = 2(7 \cdot 3 + 7 \cdot X + 3 \cdot X)

Simplify by calculating the known products:

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

Combine like terms:

102=2(21+10X) 102 = 2(21 + 10X)

Distribute the 2 across the terms inside the parentheses:

102=42+20X 102 = 42 + 20X

To isolate X X , subtract 42 from both sides:

60=20X 60 = 20X

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

X=6020=3 X = \frac{60}{20} = 3

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

3

Final Answer

3

Key Points to Remember

Essential concepts to master this topic
  • Formula: Surface area of cuboid = 2(lw + lh + wh)
  • Technique: Substitute known values: 102 = 2(7×3 + 7×X + 3×X)
  • Check: Verify X=3: 2(21 + 21 + 9) = 2(51) = 102 ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting the coefficient 2 in the surface area formula
    Don't use S = lw + lh + wh = wrong answer by half! This misses that each face appears twice on a cuboid. Always use S = 2(lw + lh + wh) to account for all six faces.

Practice Quiz

Test your knowledge with interactive questions

A cuboid is shown below:

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What is the surface area of the cuboid?

FAQ

Everything you need to know about this question

Why do we multiply by 2 in the surface area formula?

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A cuboid has 6 faces that come in 3 pairs of identical rectangles. Each pair (top/bottom, front/back, left/right) has the same area, so we calculate 3 different areas and multiply by 2.

How do I know which dimension is which?

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It doesn't matter! You can assign length, width, and height to any of the three dimensions. The surface area formula works regardless of which dimension you call l, w, or h.

What if I get a negative answer for X?

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Check your arithmetic! Dimensions must be positive numbers since they represent physical measurements. A negative result means there's an error in your calculations.

Can I solve this without the surface area formula?

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No, you need the formula! The surface area formula is the only way to relate the given surface area (102) to the unknown dimension X. Memorize it: S=2(lw+lh+wh) S = 2(lw + lh + wh)

Why do we combine like terms 7X + 3X?

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Both terms have the same variable X, so we can add their coefficients: 7X + 3X = 10X. This simplifies our equation and makes it easier to solve for X.

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