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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.
l
=
30

in
.
, w=20 in.
w
=
20

in
.
, h=18 in.
h
=
18

in
.
, and x=36.1 in.
x
=
36.1

in
.
, find the length of d
d
. Round the answer to the nearest tenth.

(1 point)

Answer 1

To find the length of \( d \) in the given three-dimensional rectangular prism, we can use the 3D distance formula. This distance represents the diagonal line from the back right vertex to the front left vertex of the prism.

Given:

• Length \( l = 30 \) in
• Width \( w = 20 \) in
• Height \( h = 18 \) in

The coordinates for the two vertices are as follows:

• Back right vertex \( (l, w, 0) = (30, 20, 0) \)
• Front left vertex \( (0, 0, h) = (0, 0, 18) \)

Now, we can apply the 3D distance formula, defined as:

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

Substituting in our points:

• \( (x_1, y_1, z_1) = (30, 20, 0) \)
• \( (x_2, y_2, z_2) = (0, 0, 18) \)

Substituting into the formula gives:

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

Calculating each term:

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

Now summing those values:

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

Calculating the square root:

\[ \sqrt{1624} \approx 40.3 \]

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

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