Calculate the Diagonal of a Rectangular Prism: Dimensions 5x, x+3, 2x+1

Q: How do I expand (x+3)² correctly?

Use the pattern $ (a+b)^2 = a^2 + 2ab + b^2 $. So $ (x+3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9 $. Don't forget the middle term!

Q: Can I factor the final expression under the square root?

Sometimes! Look for common factors first. Here, $ 30x^2 + 10x + 10 = 10(3x^2 + x + 1) $, so you get $ \sqrt{10} \cdot \sqrt{3x^2 + x + 1} $.

Q: What if x has a specific value?

If given a value for x, substitute it into $ \sqrt{30x^2 + 10x + 10} $ and calculate! For example, if x = 2, you'd get $ \sqrt{30(4) + 10(2) + 10} = \sqrt{150} $.

Q: Why can't I simplify the square root further?

The expression $ 30x^2 + 10x + 10 $ doesn't factor into a perfect square. You can factor out 10, but $ 3x^2 + x + 1 $ doesn't factor nicely over the integers.

3D Diagonal Formula with Algebraic Expressions

A rectangular prism has dimensions of $5x,x+3,2x+1$ .

Calculate the length of its diagonal.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the length of the box diagonal

00:03 Use the Pythagorean theorem in triangle C1CB to find C1B

00:16 Substitute appropriate values according to the given data and solve for BC1

00:26 Open parentheses properly

00:37 Collect terms

00:45 Draw the face diagonal

00:51 Now use the Pythagorean theorem in triangle ABC1 to find AC1

01:01 Substitute appropriate values according to the given data and solve for AC1

01:17 Collect terms

01:34 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A rectangular prism has dimensions of $5x,x+3,2x+1$ .

Calculate the length of its diagonal.

Step-by-step solution

To solve for the diagonal of the rectangular prism, we apply the three-dimensional Pythagorean theorem.

The formula for the diagonal $d$ of a rectangular prism with side lengths $a$ , $b$ , and $c$ is:

$d = \sqrt{a^2 + b^2 + c^2}$

Substituting the given dimensions into the formula, we get:

$a = 5x$ , $b = x + 3$ , $c = 2x + 1$

Therefore, the expression for the diagonal becomes:

$d = \sqrt{(5x)^2 + (x+3)^2 + (2x+1)^2}$

Calculating each squared term:

$(5x)^2 = 25x^2$
$(x+3)^2 = x^2 + 6x + 9$
$(2x+1)^2 = 4x^2 + 4x + 1$

Add these results together:

$25x^2 + x^2 + 6x + 9 + 4x^2 + 4x + 1$

Simplify the expression:

$25x^2 + x^2 + 4x^2 = 30x^2$
$6x + 4x = 10x$
$9 + 1 = 10$

Thus, the expression inside the square root becomes:

$30x^2 + 10x + 10$

Finally, the length of the diagonal is:

$d = \sqrt{30x^2 + 10x + 10}$

Therefore, the solution to the problem is $\sqrt{30x^2 + 10x + 10}$ .

Final Answer

$\sqrt{30x^2+10x+10}$

Key Points to Remember

Essential concepts to master this topic

Formula: For rectangular prism diagonal use $d = \sqrt{a^2 + b^2 + c^2}$
Technique: Expand each squared term: $(x+3)^2 = x^2 + 6x + 9$
Check: Combine like terms: $25x^2 + x^2 + 4x^2 = 30x^2$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to square all three dimensions
Don't just add the dimensions directly like 5x + (x+3) + (2x+1) = 8x+4! This ignores the Pythagorean theorem completely and gives a linear expression instead of under a square root. Always square each dimension first, then add: $(5x)^2 + (x+3)^2 + (2x+1)^2$ .

Practice Quiz

Test your knowledge with interactive questions

Consider a right-angled triangle, AB = 8 cm and AC = 6 cm.
Calculate the length of side BC.

FAQ

Everything you need to know about this question

Why do I need to use three dimensions instead of two?

A rectangular prism is 3D, so its diagonal cuts through length, width, AND height! The regular Pythagorean theorem $a^2 + b^2 = c^2$ only works for 2D shapes like rectangles.

How do I expand (x+3)² correctly?

Use the pattern $(a+b)^2 = a^2 + 2ab + b^2$ . So $(x+3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9$ . Don't forget the middle term!

Can I factor the final expression under the square root?

Sometimes! Look for common factors first. Here, $30x^2 + 10x + 10 = 10(3x^2 + x + 1)$ , so you get $\sqrt{10} \cdot \sqrt{3x^2 + x + 1}$ .

What if x has a specific value?

If given a value for x, substitute it into $\sqrt{30x^2 + 10x + 10}$ and calculate! For example, if x = 2, you'd get $\sqrt{30(4) + 10(2) + 10} = \sqrt{150}$ .

Why can't I simplify the square root further?

The expression $30x^2 + 10x + 10$ doesn't factor into a perfect square. You can factor out 10, but $3x^2 + x + 1$ doesn't factor nicely over the integers.

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