Answer :
Answer:
The number of CDs = 111.36
The number of movie videos = 242.72
N/B: I choose not to round up the answers.
Explanation:
The method used is the Lagrangian method. Basically, the optimization problem we are trying to solve is the utility function [tex]u(x,y) = 20x+80y -x^2 -y^2[/tex]
subject to the constraint
[tex]11x + 22y = 6565[/tex].
So the optimization problem(Lagrangian) is
[tex]\Delta = 20x + 80y -x^2 -y^2- \lambda(11x+22y-6565)[/tex],
where [tex]\lambda[/tex] is a constant called the Lagrange multiplier.
To find the optimal consumption, we need to maximize the Lagrangian with respect to the variables [tex]x,y,\lambda[/tex]. This we do by differentiating [tex]\Delta[/tex] with respect to each variable and then equate to 0.
[tex]\Delta_x : 11\lambda = 20 - 2x ........................(1) \\\Delta_y: 11\lambda = 40 -y .........................(2) \\\Delta_\lambda = 11x + 22y = 6565............................(3) \\[/tex]
Equate (1) and (2), to get [tex]y = 20+2x[/tex] and substitute into (3) to get [tex]x = 111.36[/tex]. Substituting [tex]x = 111.36[/tex] into [tex]20+2x[/tex] to get the corresponding value of [tex]y[/tex].