Use the image to answer the question.

Question

An illustration shows a three dimensional rectangular prism with labeled parts. The top, front and right sides are visible. The bottom, left, and rear sides are not visible, but their positions are indicated by dashed lines. The vertical edge joining the front and left sides is labeled h. A right angle is indicated where that edge meets the front base edge. The front base edge is labeled l. A right angle is indicated where that edge meets the right base edge. The right base edge is labeled w. Two dotted lines extend from the vertex at the back right side of the prism through the interior. The first dotted line extends up to the vertex at the front left side and is labeled d. The second dotted line extends diagonally across the base to the vertex at the front left side where the perpendicular height line meets the base edge and is labeled x.

Given l=30 in.
, w=20 in.
, h=18 in.
, and x=36.1 in.
, find the length of d
. Round the answer to the nearest tenth.

Answer 1

To find the length of \( d \), which represents the diagonal from the back right corner of the rectangular prism to the front left corner, we can use the 3D distance formula. The formula for the distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in a three-dimensional space is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

In this case, we can define the coordinates of the vertices as follows (assuming the front bottom left corner of the prism is at the origin \( (0, 0, 0) \)):

• The back right corner of the prism will have coordinates \( (l, w, h) = (30, 20, 18) \).
• The front left corner of the prism will have coordinates \( (0, 0, 0) \).

Now we can plug these coordinates into the distance formula:

\[ d = \sqrt{(30 - 0)^2 + (20 - 0)^2 + (18 - 0)^2} \] \[ d = \sqrt{30^2 + 20^2 + 18^2} \] \[ d = \sqrt{900 + 400 + 324} \] \[ d = \sqrt{1624} \] \[ d \approx 40.31 \]

Rounding to the nearest tenth, the length of \( d \) is:

\[ \boxed{40.3} \text{ in.} \]