16. [0/2.5 Points] DETAILS MY NOTES WANEFMAC8 6.2.027. PREVIOUS ANSWERS ASK YOUR TEACHER PRACTICE ANOTHER Nutrition Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 216 grams of protein and at least 166 grams of fat. If chicken costs $$ 9 \mathbb{4} $$, per ounce and grain costs $$ 1 \mathbb{C} $$ per ounce, how many ollnces of each should Ruff use in each bag of dog food to minimize cost? (If an answer does not exist, enter DNE.) grain $$ \square $$ 58 oz chicken 10 oz

Question

UpStudy AI · Accepted Answer

To solve this problem, we need to set up a system of inequalities based on the nutritional requirements and then formulate a cost function to minimize.

### Known Conditions:
1. **Nutritional Content:**
   - Chicken: 10 grams of protein and 5 grams of fat per ounce.
   - Grain: 2 grams of protein and 2 grams of fat per ounce.

2. **Minimum Requirements:**
   - At least 216 grams of protein.
   - At least 166 grams of fat.

3. **Costs:**
   - Chicken costs $$ 9 $$ per ounce.
   - Grain costs $$ 1 $$ per ounce.

### Variables:
Let:
- $$ x $$ = ounces of chicken
- $$ y $$ = ounces of grain

### Inequalities:
From the nutritional content, we can derive the following inequalities:

1. Protein constraint:
   $$
   10x + 2y \geq 216
   $$

2. Fat constraint:
   $$
   5x + 2y \geq 166
   $$

### Cost Function:
The cost function to minimize is:
$$
C = 9x + 1y
$$

### Steps to Solve:
1. Set up the inequalities.
2. Graph the inequalities to find the feasible region.
3. Identify the corner points of the feasible region.
4. Evaluate the cost function at each corner point to find the minimum cost.

Let's solve the inequalities to find the feasible region.

### Step 1: Solve the inequalities for $$ y $$:
1. From $$ 10x + 2y \geq 216 $$:
   $$
   2y \geq 216 - 10x \implies y \geq 108 - 5x
   $$

2. From $$ 5x + 2y \geq 166 $$:
   $$
   2y \geq 166 - 5x \implies y \geq 83 - \frac{5}{2}x
   $$

Now we have two inequalities:
1. $$ y \geq 108 - 5x $$
2. $$ y \geq 83 - \frac{5}{2}x $$

### Step 2: Find intersection points:
To find the intersection of the two lines, we set:
$$
108 - 5x = 83 - \frac{5}{2}x
$$

Let's solve this equation for $$ x $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$108-5x=83-\frac{5}{2}x$$
- step1: Multiply both sides of the equation by LCM:
  $$\left(108-5x\right)\times 2=\left(83-\frac{5}{2}x\right)\times 2$$
- step2: Calculate:
  $$216-10x=\left(83-\frac{5}{2}x\right)\times 2$$
- step3: Calculate:
  $$216-10x=166-5x$$
- step4: Move the expression to the left side:
  $$-10x+5x=166-216$$
- step5: Add and subtract:
  $$-5x=166-216$$
- step6: Add and subtract:
  $$-5x=-50$$
- step7: Change the signs:
  $$5x=50$$
- step8: Divide both sides:
  $$\frac{5x}{5}=\frac{50}{5}$$
- step9: Divide the numbers:
  $$x=10$$
The intersection point occurs at $$ x = 10 $$. Now, we can substitute this value back into one of the equations to find $$ y $$.

Using the first equation:
$$
y = 108 - 5(10) = 108 - 50 = 58
$$

So, one intersection point is $$ (10, 58) $$.

### Step 3: Find other corner points:
Next, we need to find the points where each inequality intersects the axes.

1. For $$ y = 108 - 5x $$:
   - When $$ x = 0 $$:
     $$
     y = 108
     $$
   - When $$ y = 0 $$:
     $$
     0 = 108 - 5x \implies x = 21.6
     $$

So, the intercepts are $$ (0, 108) $$ and $$ (21.6, 0) $$.

2. For $$ y = 83 - \frac{5}{2}x $$:
   - When $$ x = 0 $$:
     $$
     y = 83
     $$
   - When $$ y = 0 $$:
     $$
     0 = 83 - \frac{5}{2}x \implies x = 33.2
     $$

So, the intercepts are $$ (0, 83) $$ and $$ (33.2, 0) $$.

### Step 4: Evaluate the cost function at the corner points:
Now we have the following corner points to evaluate:
1. $$ (10, 58) $$
2. $$ (0, 108) $$
3. $$ (0, 83) $$
4. $$ (21.6, 0) $$

Let's calculate the cost $$ C = 9x + 1y $$ for each point.

1. For $$ (10, 58) $$:
   $$
   C = 9(10) + 1(58) = 90 + 58 = 148
   $$

2. For $$ (0, 108) $$:
   $$
   C = 9(0) + 1(108) = 0 + 108 = 108
   $$

3. For $$ (0, 83) $$:
   $$
   C = 9(0) + 1(83) = 0 + 83 = 83
   $$

4. For $$ (21.6, 0) $$:
   $$
   C = 9(21.6) + 1(0) = 194.4 + 0 = 194.4
   $$

### Step 5: Determine the minimum cost:
The minimum cost occurs at the point $$ (0, 83) $$ with a cost of $$ 83 $$.

### Conclusion:
To minimize cost, Ruff should use:
- **0 ounces of chicken**
- **83 ounces of grain**

If you have any further questions or need additional assistance, feel free to ask!
Ruff should use 0 ounces of chicken and 83 ounces of grain to minimize the cost.

Upstudy AI Solution

Answer

Solution

Known Conditions:

Variables:

Inequalities:

Cost Function:

Steps to Solve:

Step 1: Solve the inequalities for :

Step 2: Find intersection points:

Step 3: Find other corner points:

Step 4: Evaluate the cost function at the corner points:

Step 5: Determine the minimum cost:

Conclusion:

Bonus Knowledge

Related Questions

Latest Algebra Questions