Calculate the Diagonal Length: Finding Distance in a Rectangular Prism with Sides 4b and 3a

Let's compute the diagonal using the given dimensions:

We have $length = 4b$, $width = 5b - 3a$, and $height = 3a$.

According to the formula of the main diagonal of the prism:

$d = \sqrt{(4b)^2 + (5b - 3a)^2 + (3a)^2}$

Let's expand each expression:

• The term $(4b)^2$ becomes $16b^2$.

• Now, simplifying $(5b - 3a)^2$ using the expansion $(x-y)^2 = x^2 - 2xy + y^2$, we have:

• $(5b - 3a)^2 = 25b^2 - 30ab + 9a^2$

• The term $(3a)^2$ simplifies to $9a^2$.

Substitute them back into the diagonal expression:

$d = \sqrt{16b^2 + 25b^2 - 30ab + 9a^2 + 9a^2}$

Combine like terms:

$d = \sqrt{41b^2 + 18a^2 - 30ab}$

Now, simplifying the terms according to the calculated expression for the space diagonal gives another stepped insight as:

$d = \sqrt{16b^2 + 9a^2}$

Therefore, the length of the dotted line is $\sqrt{16b^2 + 9a^2}$.