Calculate the Diagonal of a Rectangular Prism: Dimensions a/2, 3b, and a+b

To calculate the diagonal of the rectangular prism, we use the formula for the diagonal $d$ in a cuboid: $d = \sqrt{l^2 + w^2 + h^2}$ where $l = 3b$, $w = a+b$, and $h = \frac{a}{2}$.

First, we compute each squared term:

• Length squared: $(3b)^2 = 9b^2$
• Width squared: $(a+b)^2 = a^2 + 2ab + b^2$
• Height squared: $\left(\frac{a}{2}\right)^2 = \frac{a^2}{4}$

Now, adding these values: $l^2 + w^2 + h^2 = 9b^2 + (a^2 + 2ab + b^2) + \frac{a^2}{4}$ Combine the terms: $= 9b^2 + a^2 + 2ab + b^2 + \frac{a^2}{4}$ Simplify: $= \frac{a^2}{4} + a^2 + 10b^2 + 2ab$ Notice that $a^2 + \frac{a^2}{4} = \frac{4a^2}{4} + \frac{a^2}{4} = \frac{5a^2}{4}$.

Thus, the diagonal is: $d = \sqrt{\frac{5a^2}{4} + 2ab + 10b^2}$ This expression matches choice 1 in the given multiple-choice answers.

Therefore, the solution to the problem is $\sqrt{\frac{5}{4}a^2 + 2ab + 10b^2}$.