Question

the pythagorean theorem in three dimensions
the length of the prism is 32 cm, the width is 24 cm, the height is 44 cm, and the length of the diagonal of the base, segment $bh$, is 40 cm. find the length of the diagonal of the rectangular prism, segment $be$. round the answer to the nearest tenth.
(1 point)
$\bigcirc$ 46.6 cm
$\bigcirc$ 54.4 cm
$\bigcirc$ 51.2 cm
$\bigcirc$ 59.5 cm

Explanation:

Step1: Use 3D Pythagorean theorem

The formula for the space diagonal \( BE \) of a rectangular prism is \( BE = \sqrt{\text{length}^2 + \text{width}^2 + \text{height}^2} \). We can also use the given base diagonal \( BH = 40 \, \text{cm} \) with the height, since \( BE = \sqrt{BH^2 + \text{height}^2} \).

Step2: Substitute known values

Substitute \( BH = 40 \, \text{cm} \) and height \( = 44 \, \text{cm} \):
\( BE = \sqrt{40^2 + 44^2} \)

Step3: Calculate squared terms

Compute the squares:
\( 40^2 = 1600 \), \( 44^2 = 1936 \)

Step4: Sum and take square root

Add the values, then find the square root:
\( BE = \sqrt{1600 + 1936} = \sqrt{3536} \approx 59.5 \)

Answer:

59.5 cm (Option: 59.5 cm)