Surface Area 4x² + 24x: Finding Rectangular Prism Height

Q: Why does the surface area formula have three terms?

A rectangular prism has 6 faces that come in 3 pairs of identical rectangles. Each pair contributes one term: $ 2lw $ (top/bottom), $ 2lh $ (front/back), and $ 2wh $ (left/right sides).

Q: How do I know which dimension is the height?

Look at the diagram carefully! The height is usually the vertical dimension or the one that's not labeled. In this problem, we see x and 2x are given, so h (height) is what we need to find.

Q: What if I get a different equation when I substitute?

Double-check your substitutions! Make sure $ l = x $, $ w = 2x $, and you're solving for h. Common errors include switching dimensions or forgetting to multiply by 2 for each pair of faces.

Q: Why does the x cancel out when I solve for h?

That's the beauty of this problem! When you get $ 6xh = 24x $, dividing both sides by $ 6x $ makes the x's cancel, giving h = 4 - a constant value regardless of what x equals.

Surface Area Formulas with Unknown Dimensions

The surface area of the rectangular prism in the diagram is $4x^2+24x$ .

Calculate the height of the rectangular prism.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the height of the box

00:03 We'll use the formula for calculating the surface area of a box

00:06 We'll substitute appropriate values and solve for height H

00:33 Let's simplify what we can

00:56 Let's isolate height H

01:05 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The surface area of the rectangular prism in the diagram is $4x^2+24x$ .

Calculate the height of the rectangular prism.

Step-by-step solution

To solve the problem, we utilize the surface area formula for a rectangular prism, where the given prism has a base of dimensions $x$ and $2x$ , and an unknown height $h$ . The full formula for surface area is given as:

$A = 2lw + 2lh + 2wh$

In this situation, $l = x$ , $w = 2x$ , and $h$ is our unknown. Substituting these values, the formula becomes:

$A = 2(x)(2x) + 2(x)h + 2(2x)h$

This simplifies to:

$A = 4x^2 + 2xh + 4xh$

Further simplification gives:

$A = 4x^2 + 6xh$

We are given the total surface area as $4x^2 + 24x$ . Setting this equal to our expression:

$4x^2 + 6xh = 4x^2 + 24x$

Subtract $4x^2$ from both sides:

$6xh = 24x$

We can then divide both sides by $6x$ to solve for $h$ :

$h = \frac{24x}{6x}$

This simplifies to:

$h = 4$

Thus, the height of the rectangular prism is $4$ cm.

Final Answer

$4$ cm

Key Points to Remember

Essential concepts to master this topic

Formula: Surface area = $2lw + 2lh + 2wh$ for rectangular prisms
Technique: Set formula equal to given expression: $4x^2 + 6xh = 4x^2 + 24x$
Check: Substitute h = 4 back into surface area formula to verify result ✓

Common Mistakes

Avoid these frequent errors

Forgetting to include all six faces in surface area calculation
Don't calculate just $2(x)(2x) = 4x^2$ and stop there = missing four side faces! This ignores most of the prism's surface and gives an incomplete equation. Always include all six rectangular faces: two bases plus four sides.

Practice Quiz

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Identify the correct 2D pattern of the given cuboid:

FAQ

Everything you need to know about this question

Why does the surface area formula have three terms?

A rectangular prism has 6 faces that come in 3 pairs of identical rectangles. Each pair contributes one term: $2lw$ (top/bottom), $2lh$ (front/back), and $2wh$ (left/right sides).

How do I know which dimension is the height?

Look at the diagram carefully! The height is usually the vertical dimension or the one that's not labeled. In this problem, we see x and 2x are given, so h (height) is what we need to find.

Can I solve this without using the surface area formula?

No - you must use the formula! Surface area problems require knowing that SA = $2lw + 2lh + 2wh$ . Without this formula, you can't connect the given expression to the prism's dimensions.

What if I get a different equation when I substitute?

Double-check your substitutions! Make sure $l = x$ , $w = 2x$ , and you're solving for h. Common errors include switching dimensions or forgetting to multiply by 2 for each pair of faces.

Why does the x cancel out when I solve for h?

That's the beauty of this problem! When you get $6xh = 24x$ , dividing both sides by $6x$ makes the x's cancel, giving h = 4 - a constant value regardless of what x equals.

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Surface Area 4x² + 24x: Finding Rectangular Prism Height

Surface Area Formulas with Unknown Dimensions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does the surface area formula have three terms?

How do I know which dimension is the height?

Can I solve this without using the surface area formula?

What if I get a different equation when I substitute?

Why does the x cancel out when I solve for h?

🌟 Unlock Your Math Potential

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