To find the length of \( d \) in the three-dimensional rectangular prism, we can use the three-dimensional distance formula. Given the coordinates of the vertices involved, we can apply the Pythagorean theorem in 3D space.
- Determine the coordinates of the relevant vertices based on the given dimensions:
- Let's define the vertices of the prism:
- Vertex at the front-left-bottom: \( (0, 0, 0) \)
- Vertex at the front-right-bottom: \( (l, 0, 0) = (30, 0, 0) \)
- Vertex at the back-left-bottom: \( (0, w, 0) = (0, 20, 0) \)
- Vertex at the back-right-bottom: \( (l, w, 0) = (30, 20, 0) \)
- Vertex at the front-left-top: \( (0, 0, h) = (0, 0, 18) \)
- Vertex at the front-right-top: \( (l, 0, h) = (30, 0, 18) \)
- Vertex at the back-left-top: \( (0, w, h) = (0, 20, 18) \)
- Vertex at the back-right-top: \( (l, w, h) = (30, 20, 18) \)
- Find the coordinates of the points involved in line segment \( d \):
- Point at the back-right-top: \( (30, 20, 18) \)
- Point at the front-left-bottom: \( (0, 0, 0) \)
- Use the 3D distance formula to calculate \( d \): \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] Substituting the coordinates of the points:
- \( (x_1, y_1, z_1) = (0, 0, 0) \)
- \( (x_2, y_2, z_2) = (30, 20, 18) \)
Thus, \[ d = \sqrt{(30 - 0)^2 + (20 - 0)^2 + (18 - 0)^2} \] \[ d = \sqrt{30^2 + 20^2 + 18^2} \] Calculating each term:
- \( 30^2 = 900 \)
- \( 20^2 = 400 \)
- \( 18^2 = 324 \)
Now, summing these values: \[ d = \sqrt{900 + 400 + 324} = \sqrt{1624} \]
Now, calculate \( \sqrt{1624} \): \[ \sqrt{1624} \approx 40.31 \]
Rounding to the nearest tenth gives: \[ d \approx 40.3 \text{ in.} \] Thus, the length of \( d \) is approximately 40.3 in.