Terrestrial Telescope Manufactures the Orion Telescope at Its Ohio Plant $$- 2 X _ { 1 } ^ { 2 } + 24 X _ { 1 } - 2 X _ { 2 } ^ { 2 } + 48 X _ { 2 } + 2 X _ { 1 } X _ { 2 }$$

question 25

Essay

Terrestrial Telescope manufactures the Orion telescope at its Ohio plant.It has been determined that the number of telescopes it can make weekly is a function of the amount of its $20,000 weekly budget allocated to salaries, X₁, and to equipment, X₂.When X₁ and X₂ are expressed in $1,000's, an economic study has shown that the number of telescopes produced can be expressed as the concave function $$- 2 X _ { 1 } ^ { 2 } + 24 X _ { 1 } - 2 X _ { 2 } ^ { 2 } + 48 X _ { 2 } + 2 X _ { 1 } X _ { 2 }$$ .No more than $15,000 of its weekly budget will be spent on machines.
A.Formulate a nonlinear programming model to maximize the weekly
production of Orion telescopes.
B.Write the Kuhn Tucker conditions for this problem.Are they
necessary conditions for optimality? Sufficient?
(Since the objective function is concave and the constraints are
linear, this is a convex programming model.Thus the Kuhn-Tucker
conditions will be both necessary and sufficient.Using Y₁, Y₂, Y₃, and Y₄ to represent the shadow prices