$$ \begin{array}{l} ext { A coordinate plane with an } x ext {-axis labeled, "number of roses," and } y ext { - } \ ext { axis labeled, "number of carnations," featuring the following: } \ ext { - A solid line passing through the ordered pairs, }(0,3),(5,8),(11 ext {, } \ ext { 14), and }(15,18) \ ext { - A shaded region above this line. } \ ext { - Another solid line passing through the ordered pairs, }(0,25),(5 ext {, } \ ext { 20), (11, } 14) ext { and }(15,10) ext {. } \ ext { - A shaded region below this line. }\end{array} $$ -4 roses and 10 carnations 10 roses and 5 carnations 10.5 roses and 10 carnations 5 roses and 15 carnations

Question

$$ \begin{array}{l}	ext { A coordinate plane with an } x 	ext {-axis labeled, "number of roses," and } y 	ext { - } \ 	ext { axis labeled, "number of carnations," featuring the following: } \ 	ext { - A solid line passing through the ordered pairs, }(0,3),(5,8),(11 	ext {, } \ 	ext { 14), and }(15,18) \ 	ext { - A shaded region above this line. } \ 	ext { - Another solid line passing through the ordered pairs, }(0,25),(5 	ext {, } \ 	ext { 20), (11, } 14) 	ext { and }(15,10) 	ext {. } \ 	ext { - A shaded region below this line. }\end{array} $$ -4 roses and 10 carnations 10 roses and 5 carnations 10.5 roses and 10 carnations 5 roses and 15 carnations

UpStudy AI · Accepted Answer

To determine which of the given points lie within the feasible region defined by the two constraints on the coordinate plane, let's analyze each component step-by-step.

### **1. Understanding the Constraints**

**First Line:**
- **Points:** (0,3), (5,8), (11,14), (15,18)
- **Slope Calculation:** 
  $$
  m = \frac{8 - 3}{5 - 0} = 1
  $$
  The slope remains consistent between all points, confirming the equation of the line is:
  $$
  y = x + 3
  $$
- **Shaded Region:** Above this line, which translates to:
  $$
  y \geq x + 3
  $$

**Second Line:**
- **Points:** (0,25), (5,20), (11,14), (15,10)
- **Slope Calculation:** 
  $$
  m = \frac{20 - 25}{5 - 0} = -1
  $$
  The slope remains consistent between all points, confirming the equation of the line is:
  $$
  y = -x + 25
  $$
- **Shaded Region:** Below this line, which translates to:
  $$
  y \leq -x + 25
  $$

### **2. Feasible Region**

The feasible region is where both conditions are satisfied simultaneously:
$$
y \geq x + 3 \quad \text{and} \quad y \leq -x + 25
$$

### **3. Evaluating the Given Points**

Let's evaluate each of the provided points against the constraints:

1. **(-4, 10)**
   - **Check:** 
     $$
     10 \geq (-4) + 3 \quad \Rightarrow \quad 10 \geq -1 \quad \text{(True)}
     $$
     $$
     10 \leq -(-4) + 25 \quad \Rightarrow \quad 10 \leq 29 \quad \text{(True)}
     $$
   - **Conclusion:** Technically satisfies both inequalities, but having a negative number of roses may not make practical sense in most contexts.

2. **(10, 5)**
   - **Check:** 
     $$
     5 \geq 10 + 3 \quad \Rightarrow \quad 5 \geq 13 \quad \text{(False)}
     $$
   - **Conclusion:** Does not satisfy the first inequality.

3. **(10.5, 10)**
   - **Check:** 
     $$
     10 \geq 10.5 + 3 \quad \Rightarrow \quad 10 \geq 13.5 \quad \text{(False)}
     $$
   - **Conclusion:** Does not satisfy the first inequality.

4. **(5, 15)**
   - **Check:** 
     $$
     15 \geq 5 + 3 \quad \Rightarrow \quad 15 \geq 8 \quad \text{(True)}
     $$
     $$
     15 \leq -5 + 25 \quad \Rightarrow \quad 15 \leq 20 \quad \text{(True)}
     $$
   - **Conclusion:** Satisfies both inequalities and is practical (positive number of roses and carnations).

### **4. Final Determination**

Among the provided options, **(5 roses and 15 carnations)** is the only point that clearly lies within the feasible region and makes practical sense.

### **Answer**

**5 roses and 15 carnations** is the combination that lies within the feasible region.
5 roses and 15 carnations are the correct numbers.

-4 roses and 10 carnations
10 roses and 5 carnations
10.5 roses and 10 carnations
5 roses and 15 carnations

Upstudy AI Solution

Answer

Solution

1. Understanding the Constraints

2. Feasible Region

3. Evaluating the Given Points

4. Final Determination

Answer

Beyond the Answer

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