A fitness center is trying to determine how to allocate funds for advertising. The manager decides to advertise on the radio and in the newspaper. Previous experience with such advertising leads the manager to expect responses when r adds are run on the radio and n adds appear in the newspaper. Each ad on the radio costs $13 and each newspaper ad costs $3. Using Lagrange multipliers, a student found that when $504 are budgeted for advertising, the maximum number of responses occurred from 26 radio ad and 56 newspaper ads and had Suppose the manager budgeted an additional $22 for advertising. What is the approximate change in the optimal value as a result of this change in the constraint level?
A) 2,710 additional response
B) 484 additional responses
C) 22 additional responses
D) 367 additional responses
E) 1 additional response

Question

A fitness center is trying to determine how to allocate funds for advertising. The manager decides to advertise on the radio and in the newspaper. Previous experience with such advertising leads the manager to expect  responses when r adds are run on the radio and n adds appear in the newspaper. Each ad on the radio costs $13 and each newspaper ad costs $3. Using Lagrange multipliers, a student found that when $504 are budgeted for advertising, the maximum number of responses occurred from 26 radio ad and 56 newspaper ads and had  Suppose the manager budgeted an additional $22 for advertising. What is the approximate change in the optimal value as a result of this change in the constraint level?
A) 2,710 additional response 
B) 484 additional responses 
C) 22 additional responses 
D) 367 additional responses 
E) 1 additional response

Quizplus · Accepted Answer

We can use the Lagrange multiplier approach to set up the following equations: Maximize: 11eaa894_6475_5add_a158_9d68e1b99fef_TB6775_11 Subject to: 13r + 3n = 504 + 22 (since an additional $22 is budgeted) The Lagrangian function is: L(r,n,λ) = 11eaa894_6475_5add_a158_9d68e1b99fef_TB6775_11 + λ(13r + 3n - 526) Taking partial derivatives and setting them equal to zero: ∂L/∂r = 11eaa894_6475_81ee_a158_d1b7f646455e_TB6775_11 + 13λ = 0 ∂L/∂n = 11eaa894_6475_81ee_a158_d1b7f646455e_TB6775_11 + 3λ = 0 ∂L/∂λ = 13r + 3n - 526 = 0 Solving for r and n: r = (526/13) - (3/13)λ n = (526/3) - (13/3)λ Substituting these back into the constraint equation: 13r + 3n = 504 + 22 13((526/13) - (3/13)λ) + 3((526/3) - (13/3)λ) = 526 Simplifying and solving for λ: λ = 2 Substituting this value of λ back into the expressions for r and n: r = 26 n = 56 Thus, the optimal values for the number of radio and newspaper ads are 26 and 56, respectively, when an additional $22 is budgeted. The change in the optimal value of the objective function (number of responses) can be approximated by calculating the directional derivative of the objective function at this point in the direction of the gradient vector of the constraint function (since the constraint function has changed from 13r + 3n = 504 to 13r + 3n = 526): D_vf = ∇f · v = (11eaa894_6475_81ee_a158_d1b7f646455e_TB6775_11) · <13,3> = 3741 (where v = <13,3>) Multiplying this by the magnitude of the change in the constraint level from $504 to $526: Δc = 22 We get: Δf ≈ D_vf Δc = 3741(22) = 82302 Thus, the approximate change in the optimal value (number of responses) is 82302, which is equivalent to adding about 484 additional responses. Therefore, the best choice is B.

A Fitness Center Is Trying to Determine How to Allocate