四根长都为\(2\)的直铁条,若再选两根长都为\(a\)的直铁条,使这六根铁条端点处相连能够焊接成一个对棱相等的三棱锥形的铁架,则此三棱锥体积的取值范围是\((\qquad)\)
\(\mathrm{A}.\left( 0,\dfrac{16\sqrt{3}}{27}\right]\) \(\qquad\mathrm{B}.\left(0,\dfrac{8\sqrt{3}}{27}\right]\) \(\qquad\mathrm{C}.\left( 0,\dfrac{2\sqrt{3}}{3}\right]\) \(\qquad\mathrm{D}.
\left(0,\dfrac{\sqrt{3}}{3} \right]\)
解析:
由于该三棱锥的对棱相等,因此可考虑构造如图所示长方体,
其中$$
BD=BA_1=C_1A_1=C_1D=2,DA_1=BC_1=a.$$
显然四面体\(B-DA_1C_1\)的体积始终是长方体\(ABCD-A_1B_1C_1D_1\)的\(\dfrac13\)倍,若设\(AB=x\),则$$
AD=AA_1=\sqrt{4-x^2},x\in\left(0,2\right).$$于是可得三棱锥\(B-DA_1C_1\)的体积为$$
V_{B-DA_1C_1}=\dfrac{1}{3}\cdot x\cdot (4-x^2),x\in\left(0,2\right).$$解得\(V_{B-DA_1C_1}\)的取值范围为\(\left( 0,\dfrac{16\sqrt{3}}{27}\right]\).
