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

00:08 Let's calculate the length of the box's diagonal.

00:13 First, use the Pythagorean theorem in triangle C one, C, B, to find the length of C one, B.

00:24 Next, substitute the given values, and solve for B, C one.

00:34 Make sure to open parentheses correctly.

00:45 Then, collect like terms together.

00:53 Now, draw the face diagonal.

00:59 Use the Pythagorean theorem in triangle A, B, C one, to find A, C on e.

01:09 Substitute the given values again, and solve for A, C one.

01:25 Collect terms once more.

01:42 And that's how we solve this problem, great job!

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

Look at the triangle in the diagram. How long is side AB?

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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