Calculate Cuboid Volume: Finding Volume in Terms of X with Dimensions (5+X, 4+X, 7)

Q: How do I remember the FOIL method correctly?

FOIL stands for First, Outer, Inner, Last:First: 5 × 4 = 20Outer: 5 × X = 5XInner: X × 4 = 4XLast: X × X = X²Then combine: X² + 5X + 4X + 20 = X² + 9X + 20

Q: What if X has a negative value? Will my volume be negative?

Great question! For this to represent a real cuboid, all dimensions must be positive. This means X > -4 (so 4+X > 0) and X > -5 (so 5+X > 0). In practice, X would be positive.

Q: Can I factor the final answer 7X² + 63X + 140?

Yes! You can factor out 7: $ 7(X^2 + 9X + 20) $. You could even factor further: $ 7(X + 4)(X + 5) $. Both forms are correct!

Algebraic Expansion with Variable Dimensions

Look at the following cuboid.

Express the volume of the cuboid in terms of X.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's express the box's volume using X.

00:08 We start with the formula for box volume.

00:12 It's height times length times width.

00:15 We'll substitute numbers to find the volume in terms of X, step by step.

00:33 Remember, open parentheses correctly. Multiply each term carefully.

00:58 Multiply each factor. Be sure to get every part.

01:18 And there you have it! That's how we find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following cuboid.

Express the volume of the cuboid in terms of X.

Step-by-step solution

To solve this problem, we'll begin by writing down the formula for the volume of a cuboid. The volume $V$ is given by:

V = \text{length} \times \text{width} \times \text{height}

Given dimensions are:

Length = $5 + X$
Width = $4 + X$
Height = $7$

Substituting these into the formula gives:

V = (5 + X) \times (4 + X) \times 7

First, expand the product of the first two terms:

(5 + X)(4 + X) = 5 \times 4 + 5 \times X + X \times 4 + X \times X

= 20 + 5X + 4X + X^2

= X^2 + 9X + 20

Now multiply this by the height (7):

V = (X^2 + 9X + 20) \times 7

= 7 \times X^2 + 7 \times 9X + 7 \times 20

= 7X^2 + 63X + 140

Thus, the volume of the cuboid in terms of $X$ is:

$7X^2 + 63X + 140$

Final Answer

$7x^2+63x+140$

Key Points to Remember

Essential concepts to master this topic

Volume Formula: Multiply length × width × height for cuboids
FOIL Method: (5+X)(4+X) = 20 + 5X + 4X + X² = X² + 9X + 20
Final Check: Multiply expanded result by height: 7(X² + 9X + 20) = 7X² + 63X + 140 ✓

Common Mistakes

Avoid these frequent errors

Adding dimensions instead of multiplying
Don't add (5+X) + (4+X) + 7 = 16+2X! This gives perimeter, not volume. Volume requires space, so it needs multiplication. Always multiply all three dimensions: (5+X) × (4+X) × 7.

Practice Quiz

Test your knowledge with interactive questions

A rectangular prism has a base measuring 5 units by 8 units.

The height of the prism is 12 units.

Calculate its volume.

FAQ

Everything you need to know about this question

Why do I expand (5+X)(4+X) first instead of multiplying by 7 immediately?

It's easier to work with two variables at a time! Expanding (5+X)(4+X) first gives you X² + 9X + 20, then you just multiply this entire expression by 7. This prevents mistakes.

How do I remember the FOIL method correctly?

FOIL stands for First, Outer, Inner, Last:

First: 5 × 4 = 20
Outer: 5 × X = 5X
Inner: X × 4 = 4X
Last: X × X = X²

Then combine: X² + 5X + 4X + 20 = X² + 9X + 20

What if X has a negative value? Will my volume be negative?

Great question! For this to represent a real cuboid, all dimensions must be positive. This means X > -4 (so 4+X > 0) and X > -5 (so 5+X > 0). In practice, X would be positive.

Can I factor the final answer 7X² + 63X + 140?

Yes! You can factor out 7: $7(X^2 + 9X + 20)$ . You could even factor further: $7(X + 4)(X + 5)$ . Both forms are correct!

How do I check if 7X² + 63X + 140 is right?

Pick a simple value for X, like X = 1. Your formula gives: 7(1)² + 63(1) + 140 = 210. Using dimensions directly: (5+1) × (4+1) × 7 = 6 × 5 × 7 = 210 ✓

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