jasmine runs a restaurant that sells two kinds of cookies. jasmine knows the restaurant must make at least…

jasmine runs a restaurant that sells two kinds of cookies. jasmine knows the restaurant must make at least 13 and at most 81 batches of the mint breezes. the restaurant must also make between 61 and 176 batches of the chocolate decadence. the batches of mint breezes take 16 minutes in the oven, while batches of chocolate decadence require 9 minutes in the oven. the restaurant only has 2016 minutes in the oven available. if batches of mint breezes generate $1.94 in profit, and batches of chocolate decadence generate $2.62, how many batches of the cookies should jasmine have the restaurant make to get the most profit? mint breezes: chocolate decadence: best profit: question help: video post to forum submit question

jasmine runs a restaurant that sells two kinds of cookies. jasmine knows the restaurant must make at least 13 and at most 81 batches of the mint breezes. the restaurant must also make between 61 and 176 batches of the chocolate decadence. the batches of mint breezes take 16 minutes in the oven, while batches of chocolate decadence require 9 minutes in the oven. the restaurant only has 2016 minutes in the oven available. if batches of mint breezes generate $1.94 in profit, and batches of chocolate decadence generate $2.62, how many batches of the cookies should jasmine have the restaurant make to get the most profit? mint breezes: chocolate decadence: best profit: question help: video post to forum submit question

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of batches of Mint Breezes and $y$ be the number of batches of Chocolate Decadence.

Step2: Set up constraints

The constraints are:

  • $13\leq x\leq81$
  • $61\leq y\leq176$
  • $16x + 9y\leq2016$

Step3: Set up objective - function

The profit function $P=1.94x + 2.62y$

Step4: Analyze the corner - points

  1. First, consider the intersection of $x = 13$ and $y=61$:
    • $P_1=1.94\times13+2.62\times61=25.22 + 159.82=185.04$
  2. Intersection of $x = 13$ and $y = 176$:
    • $16\times13+9\times176=208+1584 = 1792<2016$
    • $P_2=1.94\times13+2.62\times176=25.22+461.12 = 486.34$
  3. Intersection of $x = 81$ and $y = 61$:
    • $16\times81+9\times61=1296 + 549=1845<2016$
    • $P_3=1.94\times81+2.62\times61=157.14+159.82 = 316.96$
  4. Solve the system of equations $\begin{cases}16x + 9y=2016\x = 81\end{cases}$
    • Substitute $x = 81$ into $16x+9y=2016$:
      • $16\times81+9y=2016$
      • $1296+9y=2016$
      • $9y=2016 - 1296=720$
      • $y = 80$
    • $P_4=1.94\times81+2.62\times80=157.14+209.6=366.74$
  5. Solve the system of equations $\begin{cases}16x + 9y=2016\y = 61\end{cases}$
    • $16x+9\times61=2016$
    • $16x+549 = 2016$
    • $16x=2016 - 549 = 1467$
    • $x=\frac{1467}{16}\approx91.69$ (not in the range of $x$ since $x\leq81$)

Answer:

Mint Breezes: 81 Chocolate Decadence: 80 Best profit: 366.74