Answer :
Final Answer:
Using the method of Lagrange multipliers, the approximate maximum value of f(x, y) subject to the constraint[tex]\( g(x, y) = 222 \)[/tex] can be estimated as 6225 .
Explanation:
In the method of Lagrange multipliers, we seek to find the critical points of the Lagrangian function, which is the objective function [tex]( f(x, y)[/tex] minus[tex]\( \lambda \)[/tex] times the constraint function [tex]\( g(x, y) \)[/tex] . Given that [tex]\( \lambda = 25 \)[/tex] and the maximum value of[tex]( f(x, y) )[/tex] subject to [tex]\( g(x, y) = 220 \)[/tex] is ( 6200 ), we can use this information to find an approximate value for the maximum of [tex]\( f(x, y) \)[/tex] subject to [tex]\( g(x, y) = 222 \).[/tex]
Since the constraint changes from[tex]( g(x, y) = 220 ) to ( g(x, y) = 222 ),[/tex] we adjust the Lagrange multiplier [tex]\( \lambda \)[/tex] accordingly. Since [tex]\( \lambda \)[/tex] is inversely proportional to the constraint value, we can estimate that the new value of [tex]\( \lambda \)[/tex] for[tex]( g(x, y) = 222 )[/tex] is [tex]\( \frac{220}{222} \times 25 \)[/tex]. Then, we can use this updated [tex]\( \lambda \)[/tex] value to find the approximate maximum value of [tex]\( f(x, y) \)[/tex] subject to the new constraint.