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.

(1 point)
in.

Answer 1

To find the length of \( d \), we use the three-dimensional distance formula. The points we will use are:

1. The vertex at the back right side can be represented by the coordinates \( (l, w, 0) = (30, 20, 0) \).
2. The vertex at the front left side can be represented by the coordinates \( (0, 0, h) = (0, 0, 18) \).

The distance \( d \) between these two points can be calculated using the formula for the distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\):

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

Plugging in the coordinates:

\[ d = \sqrt{(0 - 30)^2 + (0 - 20)^2 + (18 - 0)^2} \]

Calculating each term:

• \( (0 - 30)^2 = 900 \)
• \( (0 - 20)^2 = 400 \)
• \( (18 - 0)^2 = 324 \)

Now summing these values:

\[ d = \sqrt{900 + 400 + 324} = \sqrt{1624} \]

Now calculating \( \sqrt{1624} \):

\[ \sqrt{1624} \approx 40.31 \]

Rounding this to the nearest tenth gives:

\[ d \approx 40.3 \text{ in.} \]

Thus, the length of \( d \) is approximately 40.3 in.