To find the length of diagonal \( d \) in the rectangular prism, we can utilize the three-dimensional distance formula. The vertices of the prism need to be identified in a three-dimensional coordinate system.
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Let's denote the vertices as follows based on the given dimensions:
- Let point A be at (0, 0, 0) - the front bottom left corner.
- Point B at (l, 0, 0) - the front bottom right corner.
- Point C at (l, w, 0) - the back bottom right corner.
- Point D at (0, w, 0) - the back bottom left corner.
- Point E at (0, 0, h) - the front top left corner.
- Point F at (l, 0, h) - the front top right corner.
- Point G at (l, w, h) - the back top right corner.
- Point H at (0, w, h) - the back top left corner.
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Using the dimensions given:
- \( l = 30 \)
- \( w = 20 \)
- \( h = 18 \)
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The coordinates for key points will therefore be:
- Point A (0, 0, 0)
- Point D (0, w, 0) which is (0, 20, 0)
- Point G (l, w, h) which is (30, 20, 18)
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The diagonal \( d \) connects point G (30, 20, 18) and point A (0, 0, 0).
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To find the length of \( d \), we can use the distance formula in three dimensions: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] Substituting in our specific points: \[ d = \sqrt{(30 - 0)^2 + (20 - 0)^2 + (18 - 0)^2} \] \[ d = \sqrt{30^2 + 20^2 + 18^2} \] \[ d = \sqrt{900 + 400 + 324} \] \[ d = \sqrt{1624} \] \[ d \approx 40.3 , \text{in.} \quad (\text{rounding to the nearest tenth}) \]
Thus, the length of \( d \) is approximately 40.3 inches.