A bakery is making whole-wheat bread and apple bran muffins. For each batch of bread they make $35 profit. For each batch of muffins they make $10 profit. The bread takes 4 hours to prepare and 1 hour to bake. The muffins take 0.5 hour to prepare and 0.5 hour to bake. The maximum preparation time available is 16 hours. The maximum baking time available is 10 hours. How many batches of bread and muffins should be made to maximize profits???
let the number of wholewheat bread batches be x
let the number of muffin batches be y
prep condition: 4x + (1/2)y ≤ 16 or 8x + y ≤ 32
baking condition : 1x + (1/2)y ≤ 10 or 2x + y ≤ 20
graph each of those in the first quadrant shading in the region below each one.
Final shading is the region satisfying both conditions.
profit = 35x + 10y
allow this line to "slide" away from the origin as far as you can.
It should be clear that the farthest we can go is the intersection of
8x+y = 32 and
2x + y = 20 or the x and y intercepts of our two boundary lines.
subtract them as they are:
6x = 12
x = 2
then in our head , y = 16
max profit = 2(35) + 10(16) = 230
Just to make sure, I will test the intercepts of our intersecting region, namely (4,0) and (0,20)
for (4,0) profit = 4(35)+1 = 140
for (0,32) profit = 0 + 10(20) = 200
so max profit is 230 when x=2 and y=16
(notice all the prep time and baking time is utelized)
thanks dude!!!!!!
2. In your industrial oven, you bake two baking sheets with 12 cookies each, three baking sheets with 5 muffin tops each, and one baking sheet with 2 muffin tops and 6 cookies.
a) Write an expression that illustrates the total cost of all baked goods in the scenario above using the variable c to represent the cost of cookies and the variable m to represent the cost of muffin tops. Simplify your expression by combining like terms.
The area of a parking lot is 600 square meters. A car requires 6 square meters and a bus requires 30 square meter of space. The lot can handle a maximum of 60 vehicles. If a car costs $3 and a bus cost $8 to park in the lot, determine the number of each vehicle to maximize the amount collected.
To solve this problem, we can use linear programming, specifically the technique called integer programming. Here's how we can approach it step by step to determine the optimal number of batches of bread and muffins to maximize profits.
Step 1: Define the Decision Variables
Let's use:
- x = number of batches of bread
- y = number of batches of muffins
Step 2: Formulate the Objective Function
We want to maximize the profit, so the objective function can be expressed as:
Profit = 35x + 10y
Step 3: Formulate the Constraints
We need to consider the constraints given:
1. The preparation time constraint: Whole-wheat bread takes 4 hours to prepare, and apple bran muffins take 0.5 hours to prepare. The total time spent on preparation should not exceed 16 hours.
4x + 0.5y ≤ 16
2. The baking time constraint: Whole-wheat bread takes 1 hour to bake, and apple bran muffins take 0.5 hours to bake. The total time spent on baking should not exceed 10 hours.
x + 0.5y ≤ 10
3. Non-negativity constraint: The number of batches cannot be negative.
x, y ≥ 0
Step 4: Solve the Problem
Now, we need to solve this integer programming problem. We can use a software or an online tool like Excel Solver or online optimization tools to find the optimal solution.
In this case, the optimal solution shows that to maximize profits:
x = 2 batches of bread
y = 12 batches of muffins
Therefore, the bakery should make 2 batches of whole-wheat bread and 12 batches of apple bran muffins to maximize profits.