A company makes two products, widgets and gadgets, which use the same materials in their production. Given a fixed amount of materials and labor, the company must decide how many widgets and gadgets to produce. If w widgets and g gadgets are produced w and g must satisfy 9w^2+4g^2=18000. The graph of this equation for w=0 g=0 is called production possibilities curve, and a point (w,g) on this curve is a production scheduel for the company. If a widget gives a profit of 3 dollars and a gadget gives a profit of 4 dollars, find the production schedule that maximizes profit, using lagrange m

Question

A company makes two products, widgets and gadgets, which use the same materials in their production. Given a fixed amount of materials and labor, the company must decide how many widgets and gadgets to produce. If w widgets and g gadgets are produced w and g must satisfy 9w^2+4g^2=18000.     The graph of this equation for w>=0 g>=0 is called production possibilities curve, and a point (w,g) on this curve is a production scheduel for the company. If a widget gives a profit of 3 dollars and a gadget gives a profit of 4 dollars, find the production schedule that maximizes profit, using lagrange m

anonymous · Answer

okk ... see my profit is 3w + 4g ... and my condition is (3w)^2 + (2g)^2 - 18000

apply  langrange we get w and g in terms of the constant $$\lambda$$ = x 

w = 3/(18x) and g = 4/(8x)  .. place these two values in the consition to get  x = 1/(120) 

hence w = 20 and g = 60 and hence my profit is  = 60 + 240 = 300