长方体三条棱长abc成等差数列,对角线长为根号14,表面积为22,则体积是多少?要计算 长方体三条棱长a

来源：百度知道 编辑：UC知道 时间：2024/05/28 10:14:54
给说下详细步骤啊。谢谢了

a = b - x, c = b + x. b > 0.

14 = a^2 + b^2 + c^2 = (b-x)^2 + b^2 + (b+x)^2 = 3b^2 + 2x^2,
22 = 2(ab + bc + ca) = 2[(b-x)b + b(b+x) + (b-x)(b+x)] = 2[3b^2 - x^2],
11 = 3b^2 - x^2,

3 = 14 - 11 = 3x^2, x = 1, 或 x = -1.
3b^2 = 11 + x^2 = 12, b^2 = 4, b = 2.
a = 1,b = 2, c = 3. 体积=abc=1*2*3=6.
或
a = 3,b = 2, c = 1.体积=abc=1*2*3=6.