To find the length of \( d \), which is the diagonal line connecting the back right vertex of the rectangular prism to the front left vertex, we can use the three-dimensional distance formula.
The formula for the distance \( d \) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
In this case, we will assume the following coordinates based on the dimensions of the prism:
- The back right vertex (bottom corner) can be defined as \( (l, w, 0) \) which is \( (30, 20, 0) \).
- The front left vertex (top corner) can be defined as \( (0, 0, h) \) which is \( (0, 0, 18) \).
Now, we can plug these values into the distance formula:
\[ d = \sqrt{(0 - 30)^2 + (0 - 20)^2 + (18 - 0)^2} \]
Calculating each part:
- \((0 - 30)^2 = (-30)^2 = 900\)
- \((0 - 20)^2 = (-20)^2 = 400\)
- \((18 - 0)^2 = 18^2 = 324\)
Now, sum these values:
\[ d = \sqrt{900 + 400 + 324} = \sqrt{1624} \]
Calculating the square root:
\[ d \approx 40.31 \]
Rounding to the nearest tenth, we find:
\[ \boxed{40.3} , \text{in.} \]