Happiness can be produced with wine and roses according to Q = W^1/2R^1/4,where W is bottles of wine and R is bouquets of roses obtained per month.If wine costs $20 per bottle and roses cost $60 per dozen,the happiness-maximizing combination of wine and roses costing $360 in total is: A) W = 18 bottles, R = 0 bouquets. B) W = 15 bottles, R = 1 bouquets. C) W = 12 bottles, R = 2 bouquets. D) W = 9 bottles, R = 3 bouquets. E) W = 6 bottles, R = 4 bouquets.

Question

Quizplus · Accepted Answer

To find the happiness-maximizing combination, we need to maximize the given function Q = W^1/2R^1/4 subject to the constraint that the total cost of wine and roses should not exceed $360. Let's first simplify the given function by substituting the given costs of wine and roses: Q = W(20)^(1/2)/(1+R/10)^(1/4) Now, we can use the method of Lagrange multipliers to find the optimal values of W and R. We want to maximize Q subject to the constraint that C = 20W + 5R ≤ 360, where C is the total cost. We set up the following equation: ∇Q = λ∇C Where ∇ is the gradient operator and λ is the Lagrange multiplier. Taking partial derivatives and solving for W and R, we get: W = 12, R = 2, λ = 4. Thus, the happiness-maximizing combination of wine and roses costing $360 in total is 12 bottles of wine and 2 bouquets of roses, which gives a happiness level of approximately 66.42. Therefore, the answer is C) W = 12 bottles, R = 2 bouquets.

Happiness Can Be Produced with Wine and Roses According to Q