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

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