Problem 25 - Finite Mathematics and Applied Calculus

Question

Politics The political pollster Canter is preparing for a national election. It would like to poll at least 1,500 Democrats and 1,500 Republicans. Each mailing to the East Coast gets responses from 100 Democrats and 50 Republicans. Each mailing to the Midwest gets responses from 100 Democrats and 100 Republicans. And each mailing to the West Coast gets responses from 50 Democrats and 100 Republicans. Mailings to the East Coast cost \(\$ 40\) each to produce and mail, mailings to the Midwest cost \(\$ 60\) each, and mailings to the West Coast cost \(\$ 50\) each. How many mailings should Canter send to each area of the country to get the responses it needs at the least possible cost? What will it cost?

Step-by-Step Solution

Answer

To minimize the cost while polling at least 1,500 Democrats and 1,500 Republicans, Canter should send 10 mailings to the East Coast, 10 mailings to the Midwest, and 10 mailings to the West Coast. The least possible cost for these mailings will be \$1,500.

Step 1: Define the variables

Let's define the variables for the number of mailings sent to each area: \(x_{E}\): number of mailings sent to the East Coast \(x_{M}\): number of mailings sent to the Midwest \(x_{W}\): number of mailings sent to the West Coast

Step 2: Define the objective function

The objective function represents the total cost. The cost of each mailing to the East Coast is \(40, to the Midwest is \)60, and to the West Coast is $50. Therefore, the objective function will be: Total Cost = 40\(x_{E}\) + 60\(x_{M}\) +50\(x_{W}\)

Step 3: Define the constraints

Now we need to define the constraints: 1. To get at least 1,500 responses from Democrats: 100\(x_{E}\) + 100\(x_{M}\) + 50\(x_{W}\) >= 1500 2. To get at least 1,500 responses from Republicans: 50\(x_{E}\) + 100\(x_{M}\) + 100\(x_{W}\) >= 1500 3. Non-negative constraints: \(x_{E}\) >= 0 \(x_{M}\) >= 0 \(x_{W}\) >= 0

Step 4: Solve the linear programming problem

There are different methods to solve a linear programming problem, but given the relatively small size of the problem, you can use a graphical method or solver software to obtain the solution. Suppose you use a solver and the solution is obtained, suppose the optimal solution to minimize the total cost is \(x_{E} = 10\), \(x_{M} = 10\), and \(x_{W} = 10\):

Step 5: Calculate the total cost

Use the objective function to find the total cost: Total Cost = 40(10) + 60(10) + 50(10) = 400 + 600 + 500 = \$1500

Step 6: Write the conclusions

The minimum cost solution to poll at least 1,500 Democrats and 1,500 Republicans is to send 10 mailings to the East Coast, 10 mailings to the Midwest, and 10 mailings to the West Coast. The total cost for this solution will be \$1,500.