MATH105 April 2012
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Question 06 (b)
According to market research, the demand curve for a local pizza restaurant satisﬁes the following relation: if p is the price of a pizza (in dollars), and q is the number of pizzas sold per day, then
The restaurant owners want to determine what price the restaurant should charge for each pizza in order to make their daily revenue as high as possible.
(b) Use the method of Lagrange multipliers to solve the problem in part (a). There is no need to justify that the solution you obtained is the absolute maximum or minimum.
A solution that does not use the method of Lagrange multipliers will receive no credit, even if the answer is correct.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Recall that the objective function is , with contraint .
Solve the constraint equation for zero to obtain h(p,q) and set up the Lagrange system
and h(p,q) = 0. Now, compute the partial derivatives of both sides with respect to the variables, p and q to set up the system of equations that need to be solved.
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We are maximizing subject to . Solving the constraint for zero we obtain , so that the Lagrange system is
Computing the gradients,
which gives us two equations, namely:
In substituting into , we get
We can factor this to find so that either
- , i.e. .
- If then , and cannot satisfy the constraint. We reject this case.
- If then . Substituting this into the constraint gives:
which mean p2 = 400, or .
We take the positive root (since price should be positive). At a price of $20 (with a demand of 10/day), the revenue is maximized.
Aside: it wouldn't make sense for either p or q to be 0 (there would be no revenue), so we can safely divide the equations (1) and (2) without the risk of dividing by 0. Or to look at it another way: if q = 0 then from the equations that forces p=0 (and vice versa), and p=q=0 could not satisfy the constraint. If we divide the equations then . Cross-multiplying gives the relation which could be used in the constraint equation to find p and q.
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