Solving for Optimal Advertising Units in Managerial Economics

[HighAllegiant]

[SOLVED] Managerial Economics Problem

SOLVED

A firm has decided through regression analysis that its sales (S) are a function of the amount of advertising (measured in units) in two different media, television (x) and magazines (y):

S(x, y) = 100 – x² + 30x – y² + 40y

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(a) Find the level of TV and magazine advertising units that maximizes the firm's sales.
(b) Suppose that the advertising budget is restricted to 31 units. Determine the level of advertising (in units) that maximizes sales subject to this budget constraint.
(c) Give an economic interpretation for the value of the Lagrangian Multiplier obtained in part (b) above.

I already solved (a) by finding the derivative with respect to (x,y) and equating to 0.
(a) X^* = 15, Y^* = 20

My prof provided the answer to (b) and I have absolutely no idea to how he arrived at it.
I assumed the the Income(M)=31 but without any given prices for (x,y), I cannot seem to apply it into a Lagrangian method.
(I'm assuming I use the Lagrangian method because it was the most recent lecture to this question. (c) also asks a follow up question which requires the Langrangian multiplier.)
Could anyone help me?

(b) X^* = 13, Y^* = 18

 

[Ackbach]

Try $x+y=31$, and plug this into your sales function, thus reducing it to a function of a single variable. The bounds on $x$, say, are $x\in[0,31]$. Use the standard Calculus I technique for maximization.

 

[HighAllegiant]

Thanks.
I just realized that I had been so used to my instructor's problems where he provides the prices for each product that I had no idea what to do when he doesn't give any prices.
(It works on problems which are similar to the one I just asked for.)