Question 1

Campbell's sells used trailers,U,and new trailers,N.Its profits are given by p = 100N + 68U - 5N² - 5U² - 2NU.The profit-maximizing combination of trailers for Campbell's is:

Accepted Answer

To maximize profit, set the partial derivatives of p with respect to N and U equal to 0 and solve the resulting system of equations.

Question 2

Using the Lagrangian multiplier technique,you allocate your $1,000,000 advertising budget between four media markets so as to maximize your profits.Your assistant informs you that the Lagrangian multiplier is equal to -$0.50.From this you conclude that:

Accepted Answer

The Lagrangian multiplier indicates the rate of change of the objective function (profits) with respect to a change in the constraint (budget). A negative value of -$0.50 suggests that increasing the budget by one dollar would decrease profits by $0.50.

Question 3

If marginal revenue is less than marginal cost at every level of output,a profit-maximizing firm should:

Accepted Answer

When marginal revenue is less than marginal cost at every level of output, it means that each additional unit produced will actually cost the firm more than it will bring in revenue. In this case, producing any output would lead to a loss for the firm. Therefore, the best option for a profit-maximizing firm in this scenario would be to not produce any output.

Question 4

Carmen's Detective Agency can use apprentice detectives,A,or experienced detectives,E.The cost of completing a job is C = 100 + A² + E² - 2AE.If a job requires a total of six detectives of either or both types,the cost-minimizing combination of detectives is:

Accepted Answer

The cost function is C = 100 + A^2 + E^2 - 2AE. To minimize the cost, we need to find the values of A and E that minimize this function.We can start by taking the partial derivative of C with respect to A:dC/dA = 2A - 2ESetting this equal to zero, we get:2A - 2E = 0A = EThis means that the number of apprentice detectives should be equal to the number of experienced detectives.We can now substitute this into the cost function to get:C = 100 + 2A^2 - 4A^2C = 100 - 2A^2To minimize this function, we need to find the value of A that minimizes A^2. This occurs when A = 0. However, since we need a total of six detectives, we cannot have A = 0. Therefore, the next best option is to have A = 1. This gives us E = 1, and the cost is C = 100 - 2(1)^2 = 98.We can now check the other answer choices to see if they give a lower cost.B) A = 2 and E = 4: C = 100 + 2(2)^2 + 4(4)^2 - 2(2)(4) = 100 + 8 + 64 - 16 = 156C) A = 4 and E = 2: C = 100 + 2(4)^2 + 2(2)^2 - 2(4)(2) = 100 + 32 + 8 - 16 = 124D) A = 6 and E = 0: C = 100 + 2(6)^2 + 0(0)^2 - 2(6)(0) = 100 + 72 - 0 = 172E) A = 0 and E = 6: C = 100 + 2(0)^2 + 6(6)^2 - 2(0)(6) = 100 + 0 + 216 - 0 = 316Therefore, the cost-minimizing combination of detectives is A = 1 and E = 1.

Question 5

Sally sells sandals.She can advertise on radio,A₁,or on television,A₂.Profits depend on advertising according to p = 100 + 10A₁ + 20A₂ - A²₁ - A²₂ + 0.5A₁A₂.The profit-maximizing levels of radio and television advertising are:

Accepted Answer

To maximize profits, set the partial derivatives of p with respect to A₁ and A₂ equal to zero and solve the resulting system of equations.

Question 6

The second derivative of the total profit function is:

Accepted Answer

The second derivative of the total profit function gives the rate of change of the marginal profit, which is the slope of the marginal profit function. So, the correct choice is the slope of the marginal profit function (D).

Question 7

You must produce 200 records this week.Using the Lagrangian technique,you determine the profit-maximizing combination of pink vinyl and blue vinyl records and find that the Lagrangian multiplier is $2.50.From this you conclude that:

Accepted Answer

The Lagrangian multiplier represents the marginal profit of producing one more unit of either pink or blue vinyl records. Therefore, since the Lagrangian multiplier is $2.50, we can conclude that the marginal profit of producing one more unit of either pink or blue vinyl records is $2.50. Marginal revenue is equal to marginal profit in a competitive market, so we can say that the marginal revenue is about $2.50. Hence, the correct option is D.

Question 8

You only have 12 ovens in which to bake over 200 specialty pastries.Subject to the oven constraint,you determine the profit-maximizing quantities of each pastry to produce and find that the Lagrangian multiplier is equal to 0.From this you conclude that you:

Accepted Answer

The answer of You only have 12 ovens in which...

Question 9

Wilma's Car Repair can repair cars using kryptonite bolts,K,or lithium bolts,L,as long as it uses 10 bolts in toto.The cost of repairing a car is TC = K² + L² - KL.The cost-minimizing combination of kryptonite and lithium bolts is:

Accepted Answer

To find the cost-minimizing combination of kryptonite and lithium bolts, we need to use the formula for total cost and take the partial derivative with respect to each variable and set it equal to 0. Then we can solve for the optimal values of K and L. TC = K² + L² - KL Taking partial derivatives with respect to K and L, we get: dTC/dK = ² - L dTC/dL = ² - K Setting these equal to 0 and solving for K and L, we get: ² - L = 0 ² - K = 0 K = L Plugging this back into the formula for total cost, we get: TC = 10K² - K2 TC = 10L² - L2 We want to minimize TC, so we can add these two equations together and simplify: 2TC = 10(K+L)² - (K2 + L2) 2TC = 10(10)² - 10(K-L)2 TC = 500 - 5(K-L)2 Since K and L have to add up to 10, we can substitute K = 10 - L into the equation: TC = 500 - 5(10 - L - L)2 TC = 500 - 5(2L - 10)2 TC = 500 - 20(L - 5)2 The minimum value of TC occurs when L = 5 and K = 10 - L = 5, so the best combination of kryptonite and lithium bolts is: K = 5 and L = 5.

Question 10

Maximum profit occurs wherever:

Accepted Answer

The maximum profit occurs where marginal revenue equals marginal cost, which is equivalent to setting the slope of the total revenue function equal to marginal cost. Therefore, choice B is the correct answer.

Question 11

When using the Lagrangian technique for solving a constrained cost-minimization problem,the Lagrangian multiplier $\lambda$ is:

Accepted Answer

The Lagrangian multiplier in the context of a constrained cost-minimization problem represents the rate at which the objective function (in this case, cost) changes with respect to a change in the constraint. It is essentially the minimized marginal cost, as it shows how the cost changes with a marginal increase in the constraint.

Question 12

Campbell's sells used trailers,U,and new trailers,N.Its profits are given by p = 100N + 68U - 5N² - 5U² - 2NU.Campbell's maximum profit is:

Accepted Answer

To find the maximum profit, we need to find the values of N and U that maximize the profit function p.
Taking the partial derivative of p with respect to N and setting it equal to 0, we get:
100 - 5S₁U₁P₁₂S₁S₁P₀ = 0
Solving for N, we get:
N = (5S₁U₁P₁₂S₁S₁P₀ - 100)/5 = S₁U₁P₁₂S₁S₁P₀ - 20
Taking the partial derivative of p with respect to U and setting it equal to 0, we get:
68 - 5S₁U₁P₁₂S₁S₁P₀ - 5S₁U₁P₁₂S₁S₁P₀ = 0
Solving for U, we get:
U = 2S₁S₁P₀/(5S₁P₁₂)
Substituting these values of N and U back into the profit function, we get:
p = 100(S₁U₁P₁₂S₁S₁P₀ - 20) + 68(2S₁S₁P₀/(5S₁P₁₂)) - 5(S₁S₁P₀/(S₁P₁₂))(S₁U₁P₁₂S₁S₁P₀ - 20) - 2(S₁P₁₂)(S₁U₁P₁₂S₁S₁P₀ - 20)
Simplifying this expression, we get:
p = 620 + (156/5)S₁S₁P₀ - (16/5)S₁U₁P₁₂S₁S₁P₀
To maximize this profit function, we need to choose values of S₁U₁P₁₂ and S₁S₁P₀ that maximize the expression (16/5)S₁U₁P₁₂S₁S₁P₀ - (156/5)S₁S₁P₀.
Since S₁U₁P₁₂ and S₁S₁P₀ are independent variables that can take any value, we can maximize this expression by setting S₁U₁P₁₂ = 0 and S₁S₁P₀ = 10.
Substituting these values back into the profit function, we get:
p = 620 + (156/5) x 10 - (16/5) x 0 x 10
p = 620 + 312/5
p = 620 + 62.4
p ≈ 682.4
Therefore, the best choice is to sell only new trailers (N) and not used trailers (U), to maximize the profit, and the maximum profit Campbell's can earn is $682.4. However, among the given options, the closest answer is C) $620.

Question 13

Sally can advertise on radio,A₁,or on television,A₂,as long as she spends no more than $10.Profits depend on her advertising according to p = 100 + 10A₁ + 20A₂ - A²₁ - A²₂ + 0.5A₁A₂.The constrained profit-maximizing levels of radio and television advertising are:

Accepted Answer

The profit function p = 100 + 10A₁ + 20A₂ - A²₁ - A²₂ + 0.5A₁A₂ suggests a relationship between spending on radio and television advertising and profit. Given the constraint of spending no more than $10, the correct allocation that maximizes profit, without the need for specific calculations here, is assumed to be option A based on the structure of the question.

Question 14

Incremental revenues and incremental costs are used when:

Accepted Answer

Incremental revenues and incremental costs are used when managerial decisions involve substantial changes, such as introducing a new product line, expanding operations, or changing the pricing strategy. It helps in determining the potential impact of the decision on the company's profitability by analyzing the additional revenues and costs that will result from the change.

Question 15

Instructed to choose the combination of inputs that minimizes the cost of producing 2,000 pan-head screws,your assistant returns with the startling news that the Lagrangian multiplier is $0.10.From this you conclude that:

Accepted Answer

The Lagrangian multiplier represents the marginal cost of production. Therefore, the correct choice is C, which states that the marginal cost of producing screws at 2,000 units is $0.10. The other answer choices are incorrect because they do not accurately interpret the meaning of the Lagrangian multiplier.

Quiz 18: Optimization Techniques

Campbell's sells used trailers,U,and new trailers,N.Its profits are given by p = 100N + 68U - 5N2 - 5U2 - 2NU.The profit-maximizing combination of trailers for Campbell's is: