I am studying for my exams and got very very stuck at a word problem on the Lagrange Methods, my biggest difficulty is to properly identify the function to be maximized (in this case) and so its constraint.
Problem. "A computer company has a monthly advertising budget of 20.000 dolars. its marketing department estimates that if x dolars are spent each month on advertising in newspapers and y dollars per month on advertising in television, then the monthy sales will be given as $$S = 80x^{1/4} y^{3/4}$$ dollars. If the profit is 10% of sales less the advertising cost, determine how to allocate the advertising budget in order to maximize the monthly profit."
- should I assume that the variables x and y from the cost function $$x+y= 20.000$$ are the same as the ones from sales? I would think that sales is a function in terms of quantity of product not of advertising, but the solution seens so simplistic that I tried as so.
- I did then $$L(x,y,\lambda) = (80x^{1/4} y^{3/4})*0.1 - 20.000 - \lambda *(x+y-20.000)$$
- I also tried substituting the cost function in terms of one variable into the "Profit" but nothing seens to work. Book solution is given as x = 5.000 and y = 15.000
Thank you for any tip on how to find equations in word problem, this being an example.
Your cost function can be simply
$$f(x, y) = x^{1/4} y^{3/4}$$
since the profit,
$$p(x, y) = 20 - 8 f(x, y)$$ (that is 10% of sales less than the advert cost of 20).
Formulate the problem as
$$\arg \underset{x, y}{\max} \, x^{1/4} y^{3/4}$$
Subject to: $x + y = 20$.
Therefore, $$L(x, y, \lambda) = x^{1/4} y^{3/4} - \lambda (x + y - 20)$$
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