Find the values of $$ x \geq 0 $$ and $$ y \geq 0 $$ that maximize $$ z=11 x+14 y $$, subject to each of the following sets of constraints. $$ \begin{array}{lll} ext { (a) } \begin{array}{rrr}x+y \leq 16 & ext { (b) } 2 x+y \leq 10 & ext { (c) } x+2 y \geq 8 \ x+5 y \leq 20 & x+5 y \leq 15 & \begin{array}{r}2 x+y \geq 10 \ x-y \leq 6\end{array} \ x+\end{array} \ ext { A. The maximum value is } \square ext { and occurs at the point } \ ext { (Simplify your answers.) } \ ext { B. There is no maximum value. }\end{array} $$

Question

Find the values of $$ x \geq 0 $$ and $$ y \geq 0 $$ that maximize $$ z=11 x+14 y $$, subject to each of the following sets of constraints. $$ \begin{array}{lll}	ext { (a) } \begin{array}{rrr}x+y \leq 16 & 	ext { (b) } 2 x+y \leq 10 & 	ext { (c) } x+2 y \geq 8 \ x+5 y \leq 20 & x+5 y \leq 15 & \begin{array}{r}2 x+y \geq 10 \ x-y \leq 6\end{array} \ x+\end{array} \ 	ext { A. The maximum value is } \square 	ext { and occurs at the point } \ 	ext { (Simplify your answers.) } \ 	ext { B. There is no maximum value. }\end{array} $$

UpStudy AI · Accepted Answer

To solve the problem of maximizing $$ z = 11x + 14y $$ under the given constraints, we will analyze each set of constraints separately. We will find the feasible region defined by the constraints and then evaluate the objective function at the vertices of this region.

### Part (a)
**Constraints:**
1. $$ x + y \leq 16 $$
2. $$ x + 5y \leq 20 $$
3. $$ x \geq 0 $$
4. $$ y \geq 0 $$

**Step 1: Graph the constraints**
- From $$ x + y \leq 16 $$, we can express $$ y $$ as $$ y \leq 16 - x $$.
- From $$ x + 5y \leq 20 $$, we can express $$ y $$ as $$ y \leq \frac{20 - x}{5} $$.

**Step 2: Find the intersection points**
- Set $$ x + y = 16 $$ and $$ x + 5y = 20 $$:
  $$
  y = 16 - x \
  x + 5(16 - x) = 20 \
  x + 80 - 5x = 20 \
  -4x = -60 \
  x = 15 \
  y = 1
  $$
- The intersection points are:
  1. $$ (0, 0) $$
  2. $$ (0, 4) $$ from $$ x + 5y = 20 $$
  3. $$ (16, 0) $$ from $$ x + y = 16 $$
  4. $$ (15, 1) $$ from the intersection above.

**Step 3: Evaluate $$ z $$ at the vertices**
- $$ z(0, 0) = 0 $$
- $$ z(0, 4) = 56 $$
- $$ z(16, 0) = 176 $$
- $$ z(15, 1) = 179 $$

**Conclusion for Part (a):**
The maximum value is $$ 179 $$ at the point $$ (15, 1) $$.

### Part (b)
**Constraints:**
1. $$ 2x + y \leq 10 $$
2. $$ x + 5y \leq 15 $$
3. $$ x \geq 0 $$
4. $$ y \geq 0 $$

**Step 1: Graph the constraints**
- From $$ 2x + y \leq 10 $$, we can express $$ y $$ as $$ y \leq 10 - 2x $$.
- From $$ x + 5y \leq 15 $$, we can express $$ y $$ as $$ y \leq \frac{15 - x}{5} $$.

**Step 2: Find the intersection points**
- Set $$ 2x + y = 10 $$ and $$ x + 5y = 15 $$:
  $$
  y = 10 - 2x \
  x + 5(10 - 2x) = 15 \
  x + 50 - 10x = 15 \
  -9x = -35 \
  x = \frac{35}{9} \
  y = 10 - 2\left(\frac{35}{9}\right) = \frac{10}{9}
  $$
- The intersection points are:
  1. $$ (0, 0) $$
  2. $$ (0, 10) $$ from $$ 2x + y = 10 $$
  3. $$ (5, 0) $$ from $$ x + 5y = 15 $$
  4. $$ \left(\frac{35}{9}, \frac{10}{9}\right) $$

**Step 3: Evaluate $$ z $$ at the vertices**
- $$ z(0, 0) = 0 $$
- $$ z(0, 10) = 140 $$
- $$ z(5, 0) = 55 $$
- $$ z\left(\frac{35}{9}, \frac{10}{9}\right) = 11\left(\frac{35}{9}\right) + 14\left(\frac{10}{9}\right) = \frac{385 + 140}{9} = \frac{525}{9} \approx 58.33 $$

**Conclusion for Part (b):**
The maximum value is $$ 140 $$ at the point $$ (0, 10) $$.

### Part (c)
**Constraints:**
1. $$ x + 2y \geq 8 $$
2. $$ 2x + y \geq 10 $$
3. $$ x - y \leq 6 $$
4. $$ x + 5y \leq 15 $$
5. $$ x \geq 0 $$
6. $$ y \geq 0 $$

**Step 1: Graph the constraints**
- From $$ x + 2y \geq 8 $$, we can express $$ y $$ as $$ y \geq 4 - \frac{x}{2} $$.
- From $$ 2x + y \geq 10 $$, we can express $$ y $$ as $$ y \geq 10 - 2x $$.
- From $$ x - y \leq 6 $$, we can express $$ y $$ as $$ y \geq x - 6 $$.
- From $$ x + 5y \leq 15 $$, we can express $$ y $$ as $$ y \leq \frac{15 - x}{5} $$.

**Step 2: Find the intersection points**
- Set $$ x + 2y = 8 $$ and $$ 2x + y = 10 $$:
  $$
  y = 8 - x \
  2x + (8 - x) = 10 \
  x + 8 = 10 \
  x = 2 \
  y = 6
  $$
- The intersection points are:
  1. $$ (0, 0) $$
  2. $$ (2, 6) $$
  3. $$ (6, 0) $$ from $$ x - y = 6 $$
  4. $$ (0, 3) $$ from $$ x + 5y = 15 $$

**Step 3: Evaluate $$ z $$ at the vertices**
- $$ z(0, 0) = 0 $$
- $$ z(2, 6) = 92 $$
- $$ z(6, 0) = 66 $$
- $$ z(0, 3) = 42 $$

**Conclusion for Part (c):**
The
**Solutions:** - **Part (a):** - **Maximum Value:** 179 - **Point:** (15, 1) - **Part (b):** - **Maximum Value:** 140 - **Point:** (0, 10) - **Part (c):** - **Maximum Value:** 92 - **Point:** (2, 6)

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