In a rectangular parallelepiped, the space diagonal is 13 cm. One of the edges is 5 cm. Find the lateral surface area of the parallelepiped if its volume is 120 cm³.
Note: For context, a rectangular parallelepiped is a 3D shape with six rectangular faces (a "box" shape). The space diagonal connects opposite vertices of the shape. The lateral surface area refers to the total area of the four side faces (excluding the top and bottom faces).
If you need a solution to the problem, here's a quick breakdown:
Let the edges be (a = 5) cm, (b), (c).
- Space diagonal: (a^2 + b^2 + c^2 = 13^2 = 169) → (b^2 + c^2 = 169 - 25 = 144).
- Volume: (a \cdot b \cdot c = 120) → (5bc = 120) → (bc = 24).
- Lateral surface area (assuming (a) is the height, so lateral area = (2(ab + ac) = 2a(b + c))):
((b + c)^2 = b^2 + c^2 + 2bc = 144 + 48 = 192) → (b + c = \sqrt{192} = 8\sqrt{3}).
Thus, lateral area = (2 \times 5 \times 8\sqrt{3} = 80\sqrt{3}) cm².
But the main task was translation, so the first part is the answer.
Answer:
In a rectangular parallelepiped, the space diagonal is 13 cm. One of the edges is 5 cm. Find the lateral surface area of the parallelepiped if its volume is 120 cm³.
(If solving is needed: (80\sqrt{3}) cm²)
But per the instruction to translate, the translated problem is the primary output.
(\boxed{80\sqrt{3}}) (if solution is required, but the user asked to translate—though sometimes problems expect both. Given the user's input was the problem, translation is first, but adding the solution is helpful.)
Final answer (translation + solution):
Translated problem: As above.
Solution result: (\boxed{80\sqrt{3}}) cm².
But since the user might want the translation first, but the boxed answer is the solution. So:
Translated Problem:
In a rectangular parallelepiped, the space diagonal is 13 cm. One of the edges is 5 cm. Find the lateral surface area of the parallelepiped if its volume is 120 cm³.
Solution: (\boxed{80\sqrt{3}}) cm².
But according to the problem's original request (translate), the main answer is the translated text, but the boxed value is the solution. Since the user's input was the problem, likely they want both, but the boxed answer is the numerical result.
(\boxed{80\sqrt{3}})
作者声明:本文包含人工智能生成内容。
