To maximize $P = 8x + 15y$ subject to the constraints:

$5x + y \leq 10$ $x + 5y \leq 15$ $x \geq 0$ $y \geq 0$

we can use the method of linear programming to find the optimal values of $x$ and $y$.

Step 1: Identify the feasible region

The constraints define the region in which $x$ and $y$ can vary. Let’s examine each inequality:

1. Constraint 1: $5x + y \leq 10$

   • Rewrite as $y \leq -5x + 10$, which is a line with slope -5 and y-intercept 10.

2. Constraint 2: $x + 5y \leq 15$

   • Rewrite as $y \leq -\frac{1}{5}x + 3$, which is a line with slope $-\frac{1}{5}$ and y-intercept 3.

3. Non-negativity constraints: $x \geq 0$ and $y \geq 0$

   • These constrain the solution to the first quadrant.

Step 2: Find the corner points

The feasible region is bounded by the intersection of the lines defined by these constraints. We calculate the intersections:

1. Intersection of $5x + y = 10$ and $x + 5y = 15$:

   • Solve these equations simultaneously.
   • Multiply the second equation by 5 to eliminate $y$: $5x + y = 10$ $5x + 25y = 75$
     • Subtract the first equation from the second: $24y = 65 \Rightarrow y = \frac{65}{24} \approx 2.71$
     • Substitute $y \approx 2.71$ into the first equation: $5x + 2.71 = 10 \Rightarrow 5x \approx 7.29 \Rightarrow x \approx 1.46$

   So, one vertex is approximately $(1.46, 2.71)$.

2. Intersection of $5x + y = 10$ and $y = 0$:

   • Set $y = 0$ in $5x + y = 10$: $5x = 10 \Rightarrow x = 2$
   • This gives the point $(2, 0)$.

3. Intersection of $x + 5y = 15$ and $x = 0$:

   • Set $x = 0$ in $x + 5y = 15$: $5y = 15 \Rightarrow y = 3$
   • This gives the point $(0, 3)$.

The vertices of the feasible region are approximately $(1.46, 2.71)$, $(2, 0)$, and $(0, 3)$.

Step 3: Evaluate $P = 8x + 15y$ at each vertex

1. At $(1.46, 2.71)$: $P = 8(1.46) + 15(2.71) \approx 11.68 + 40.65 = 52.33$

2. At $(2, 0)$: $P = 8(2) + 15(0) = 16$

3. At $(0, 3)$: $P = 8(0) + 15(3) = 45$

Conclusion

The maximum value of $P$ occurs at $(1.46, 2.71)$, where $P \approx 52.33$.

Let me know if you'd like further details or have any questions!

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1. How do you graph the constraints to visualize the feasible region?
2. Why is the feasible region bounded by the constraints?
3. What is the purpose of checking each vertex in linear programming?
4. How would this problem change if an additional constraint was added?
5. How does the slope of each constraint line affect the feasible region?

Tip: When solving linear programming problems, always check the vertices of the feasible region for optimal solutions.