To find the length of \( d \), we can use the three-dimensional distance formula. In this case, \( d \) represents the distance between the vertex at the back right side of the prism and the vertex at the front left side.
The coordinates of the vertices can be defined as follows:
- The back right vertex \( (l, w, h) = (30, 20, 18) \)
- The front left vertex \( (0, 0, 0) \)
Using the distance formula for 3D space, the distance \( d \) can be calculated with the following formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Given:
- \( x_1 = 0, y_1 = 0, z_1 = 0 \) (front left vertex)
- \( x_2 = 30, y_2 = 20, z_2 = 18 \) (back right vertex)
Substituting the coordinates into the formula:
\[ d = \sqrt{(30 - 0)^2 + (20 - 0)^2 + (18 - 0)^2} \] \[ d = \sqrt{30^2 + 20^2 + 18^2} \] \[ d = \sqrt{900 + 400 + 324} \] \[ d = \sqrt{1624} \] Calculating the square root:
\[ d \approx 40.31 \]
Rounding to the nearest tenth gives:
\[ d \approx 40.3 \text{ in.} \]
So, the length of \( d \) is approximately 40.3 in.