two bakeries produce three types of cookies, chocolate chip, sugar, and peanut butter. the cookies are produced in batches containing some of each type of cookie. the cooks will not break up a batch, so you must buy an entire batch of cookies if you want any cookies. mrs. betty's bakery produces 3 dozen CC, 4 dozen S, and 1 dozen PB in a batch, and it charges 30 for the cookies produced in a batch. crazy cookies produces 5 dozen CC, 1 dozen S, and 2 dozen PB in a batch, and it charges 35 dollars for the cookies produced in a batch. if a consumer wants t least 100 dozen CC, 35 dozen S and 14 dozen PB, how many batches should be ordered from each bakery in order to minimize costs.
To minimize costs, we need to find the combination of batches from each bakery that satisfies the given requirements and has the lowest total cost.
Let's start by defining the variables:
Let x be the number of batches ordered from Mrs. Betty's bakery.
Let y be the number of batches ordered from Crazy Cookies bakery.
Now, let's calculate the number of dozens of each cookie type provided by each bakery per batch:
Mrs. Betty's Bakery:
- Chocolate Chip (CC): 3 dozen per batch
- Sugar (S): 4 dozen per batch
- Peanut Butter (PB): 1 dozen per batch
Crazy Cookies Bakery:
- Chocolate Chip (CC): 5 dozen per batch
- Sugar (S): 1 dozen per batch
- Peanut Butter (PB): 2 dozen per batch
Next, we can calculate the total number of dozens of each cookie type obtained from each bakery based on the number of batches ordered:
Mrs. Betty's Bakery:
- Number of dozens of CC = 3 * x
- Number of dozens of S = 4 * x
- Number of dozens of PB = 1 * x
Crazy Cookies Bakery:
- Number of dozens of CC = 5 * y
- Number of dozens of S = 1 * y
- Number of dozens of PB = 2 * y
According to the given requirements, we need at least:
- 100 dozen CC
- 35 dozen S
- 14 dozen PB
We can now write the following inequality equations based on the required number of dozens for each cookie type:
1) 3x + 5y ≥ 100 (equation for CC)
2) 4x + y ≥ 35 (equation for S)
3) x + 2y ≥ 14 (equation for PB)
To minimize the cost, we need to minimize the total cost function, which can be calculated as follows:
Cost = 30x + 35y
Now, we have a system of inequality equations and a cost function. We can solve this using linear programming techniques, such as the simplex method or graphical method, to find the optimal values of x and y that minimize the cost while satisfying the given requirements. These techniques are typically solved using specialized software or spreadsheet applications (such as Microsoft Excel).
Once the optimal values of x and y are obtained, they will represent the number of batches to be ordered from each bakery to minimize costs.