An online dating service is trying to determine how to allocate funds for advertising. The manager decides to advertise on the radio and through internet ads. Previous experience with such advertising leads the dating service to expect f(r, n) = 0.8r²n responses when r ads are run on the radio and n internet ads are run. Check: (10,30)=2400 Each ad on the radio costs $140, and each internet ad costs $70 and the dating service has budgeted $9240 for advertising, so the advertising constraint is g(r, n) = 140r+70n=9240 dollars. (a) Choose the system of equations that could be used to find the optimal number of responses subject to the advertising constraint. 1.6mm + 1402 = 0 1.6mm = A 0.8²2²=A 1.6mm = 0 0.8r² = 0 1.6mm = 140 0.8r² = 70 O 0.8² +702=0 140r +70n 9240 140r+ 70n9240 140r + 70n = 9240 X 140r + 70n9240 You do NOT need to solve the system. The solution is provided. The optimal point subject to the constraint was found to occur where: r=44, n=44, λ=22.126, and f=68147.2.
An online dating service is trying to determine how to allocate funds for advertising. The manager decides to advertise on the radio and through internet ads. Previous experience with such advertising leads the dating service to expect f(r, n) = 0.8r²n responses when r ads are run on the radio and n internet ads are run. Check: (10,30)=2400 Each ad on the radio costs $140, and each internet ad costs $70 and the dating service has budgeted $9240 for advertising, so the advertising constraint is g(r, n) = 140r+70n=9240 dollars. (a) Choose the system of equations that could be used to find the optimal number of responses subject to the advertising constraint. 1.6mm + 1402 = 0 1.6mm = A 0.8²2²=A 1.6mm = 0 0.8r² = 0 1.6mm = 140 0.8r² = 70 O 0.8² +702=0 140r +70n 9240 140r+ 70n9240 140r + 70n = 9240 X 140r + 70n9240 You do NOT need to solve the system. The solution is provided. The optimal point subject to the constraint was found to occur where: r=44, n=44, λ=22.126, and f=68147.2.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.4: Definition Of The Derivative
Problem 50E
Related questions
Question
![### Advertising Budget Allocation for an Online Dating Service
An online dating service is determining how to allocate funds for advertising effectively. The manager plans to advertise on the radio and through internet ads, leveraging previous data to estimate responses.
#### Mathematical Model
The expected number of responses \( f(r, n) \) when \( r \) radio ads and \( n \) internet ads are run is given by:
\[ f(r, n) = 0.8r^2 + n \]
For instance, if 10 radio ads and 30 internet ads are run:
\[ f(10, 30) = 2400 \]
Each radio ad costs $140, each internet ad costs $70, and the total budget is $9240:
\[ g(r, n) = 140r + 70n = 9240 \]
#### Problem Formulation
**(a) Choose the system of equations to find the optimal number of responses subject to the budget constraint.**
Given options:
1. \(1.6r = \lambda \)
\[ 140r + 70n = 9240 \]
2. \(1.6r + 140 \lambda = 0 \)
\[ 140r + 70n = 9240 \]
3. \(0.8r^2 = \lambda \)
\[ 140r + 70n = 9240 \]
4. \(0.8r^2 + 70\lambda = 0 \]
5. \(0.8r^2 = 704 \)
The correct choice is:
\[ 0.8r^2 = \lambda \]
\[ 140r + 70n = 9240 \]
#### Solution
You do not need to solve the system. The solution is provided.
**The optimal point subject to the constraint occurs at:**
\[
r = 44, n = 44, \lambda = 22.126, f = 68147.2
\]
**(b) Use the close-point method to classify the optimal point.** (Do NOT round.)
| | \( r \) | \( n \) | \( f(r,n) \) |
|--------|---------|---------|---------------|
| Close point 1 | 43 | 44 | 68147.2 (incorrect) |
| Optimal point | 44](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcfa3f98-eb44-4f4d-86d7-8c297108b84f%2Ffaabbb98-93c7-425a-980d-f2c66da68326%2Fwbl180v_processed.png&w=3840&q=75)
Transcribed Image Text:### Advertising Budget Allocation for an Online Dating Service
An online dating service is determining how to allocate funds for advertising effectively. The manager plans to advertise on the radio and through internet ads, leveraging previous data to estimate responses.
#### Mathematical Model
The expected number of responses \( f(r, n) \) when \( r \) radio ads and \( n \) internet ads are run is given by:
\[ f(r, n) = 0.8r^2 + n \]
For instance, if 10 radio ads and 30 internet ads are run:
\[ f(10, 30) = 2400 \]
Each radio ad costs $140, each internet ad costs $70, and the total budget is $9240:
\[ g(r, n) = 140r + 70n = 9240 \]
#### Problem Formulation
**(a) Choose the system of equations to find the optimal number of responses subject to the budget constraint.**
Given options:
1. \(1.6r = \lambda \)
\[ 140r + 70n = 9240 \]
2. \(1.6r + 140 \lambda = 0 \)
\[ 140r + 70n = 9240 \]
3. \(0.8r^2 = \lambda \)
\[ 140r + 70n = 9240 \]
4. \(0.8r^2 + 70\lambda = 0 \]
5. \(0.8r^2 = 704 \)
The correct choice is:
\[ 0.8r^2 = \lambda \]
\[ 140r + 70n = 9240 \]
#### Solution
You do not need to solve the system. The solution is provided.
**The optimal point subject to the constraint occurs at:**
\[
r = 44, n = 44, \lambda = 22.126, f = 68147.2
\]
**(b) Use the close-point method to classify the optimal point.** (Do NOT round.)
| | \( r \) | \( n \) | \( f(r,n) \) |
|--------|---------|---------|---------------|
| Close point 1 | 43 | 44 | 68147.2 (incorrect) |
| Optimal point | 44
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,