Use the image to answer the question.
An illustration of a rectangular prism is shown with labeled parts. The left and back base edges are not visible but their positions are indicated by a dashed line. The 6 vertices are labeled A B C D E F and G. The base edge A B is labeled length. Base edge B C is labeled width. The perpendicular C D is labeled height. A dotted line crosses the rectangular base through the center to show diagonal B H. A dashed line extends up from the base indicating the left rear perpendicular E H. A dotted line crosses from the back top left vertex E to the front bottom right vertex B to show diagonal B E. This dotted line is labeled diagonal.
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)
Responses
59.5 cm
59.5 cm
51.2 cm
51.2 cm
54.4 cm
54.4 cm
46.6 cm
To find the length of the diagonal of the rectangular prism, we can use the Pythagorean theorem.
In the triangle BHE, BH is given as 40 cm, and BE is the diagonal we want to find. We can use the length and height to find the length of HE.
Applying the Pythagorean theorem, we get:
HE^2 = BH^2 - BE^2
HE^2 = (32 cm)^2 - (24 cm)^2
HE^2 = 1024 cm^2 - 576 cm^2
HE^2 = 448 cm^2
Taking the square root of both sides, we find:
HE = √448 cm
HE ≈ 21.2 cm
Now, in the triangle BEH, we have the lengths BE and HE. Again, applying the Pythagorean theorem:
BE^2 = BH^2 + HE^2
BE^2 = (40 cm)^2 + (21.2 cm)^2
BE^2 = 1600 cm^2 + 449.44 cm^2
BE^2 = 2049.44 cm^2
Taking the square root of both sides, we get:
BE = √2049.44 cm
BE ≈ 45.3 cm
Rounding to the nearest tenth, the length of the diagonal of the rectangular prism, segment BE, is approximately 45.3 cm. Therefore, the correct answer is:
45.3 cm