In some parts of the world, Red Lizzard Wine is alleged to increase one's longevity.It is produced by the process Q  min(1/3) L, R, where L is the number of spotted red lizards and R is gallons of rice wine.P_L  P_R  $1.Demand for Red Lizzard Wine in the United States is Q  576P^2 A^1/2.If the advertising budget is $121, the quantity of wine which should be imported into the United States is

A) 0 gallons.
B) 33 gallons.
C) 396 gallons.
D) 99 gallons.
E) 95 gallons.

To find the optimal quantity to import, we need to maximize profit, which is equal to total revenue minus total cost. Total revenue can be found by multiplying the demand equation by the price:

TR = P*Q = $1*Q = Q

Total cost is the cost of producing the wine (which we assume is constant at all levels of production) plus the cost of importing:

TC = 121 + R

Putting these together, we get:

Profit = TR – TC = Q – (121 + R)

To maximize profit, we take the derivative with respect to R and set it equal to zero:

dProfit/dR = -1

So we want to set the cost of importing (R) equal to the difference between the optimal quantity (Q) and the advertising budget (121):

R = Q - 121

Now we just need to plug in the given equations for demand and production to solve for Q.

From the production equation, we have:

R = 99 gallons

Substituting this into the demand equation, we get:

Q = 576 – 33P

Substituting P = 1, we get:

Q = 543

So the optimal quantity of wine to import is 543 – 121 = 422.

However, the production equation only allows for integer values of R (since you can't import a fraction of a gallon of wine). The closest integer value to 99 that satisfies the equation is R = 95. Substituting this into the demand equation gives:

Q = 576 – 33P

Substituting P = 1, we get:

Q = 543

So the optimal quantity of wine to import is 543 – 121 = 422, which is between 396 and 495 (the answer choices closest to 422). Since we can only import integer values of R, the closest option is D) 99 gallons.