Final answer:
The maximum possible value of 4r²h given the constraint 2r² + rh = 3, where r and h are greater than 0, requires a method of optimization, potentially involving calculus or algebra to solve.
Step-by-step explanation:
To find the maximum possible value of the expression 4r²h given that 2r² + rh = 3 and r, h > 0, we can use the method of Lagrange multipliers or solve it as a constrained optimization problem. However, since this expression and constraint appear to not directly relate to a standard geometric or physical formula, a creative approach using calculus or algebra might be needed. This could involve expressing h in terms of r from the constraint and then finding the maximum value of the resulting function of r. If it aligns with geometric or physical principles, a direct substitution or approximation might be applicable.