Chapter 16, Problem 20P

A cylindrical beam carries a compression load $P=3000\text{ kN}$. To prevent the beam from buckling, this load must be less than a critical load,

$P_c=\frac{{\pi }^{2}EI}{L^2}$

where $E=\text{Young's modulus}=200\times {10}^9{\text{N/m}}^2,I=\pi r^4/4$(the area moment of inertia for a cylindrical beam of radius r), and L is the beam length. If the volume of beam V cannot exceed. $0.075{\text{ m}}^{\text{3}}$, find the largest height L that can be utilized and the corresponding radius.