To determine which of the points maximizes John's profit, we need to evaluate each point given against the constraints and then calculate the profit for each point that satisfies all the constraints.
The constraints are:
1. [tex]\( s \leq 40 - 3t \)[/tex]
2. [tex]\( s \geq 12 - t \)[/tex]
3. [tex]\( s \leq 30 - t \)[/tex]
4. [tex]\( s \geq 0 \)[/tex]
5. [tex]\( t \geq 0 \)[/tex]
And the profit function is given by:
[tex]\[ \text{Profit} = 14s + 6t \][/tex]
Let's evaluate the points one by one.
### Point A: [tex]\((0, 13.33)\)[/tex]
- Checking the constraints:
1. [tex]\( 0 \leq 40 - 3(13.33) \Rightarrow 0 \leq 40 - 39.99 \Rightarrow 0 \leq 0.01 \)[/tex] (True)
2. [tex]\( 0 \geq 12 - 13.33 \Rightarrow 0 \geq -1.33 \)[/tex] (True)
3. [tex]\( 0 \leq 30 - 13.33 \Rightarrow 0 \leq 16.67 \)[/tex] (True)
4. [tex]\( 0 \geq 0 \)[/tex] (True)
5. [tex]\( 13.33 \geq 0 \)[/tex] (True)
- All constraints are satisfied.
- Calculating the profit:
[tex]\[ \text{Profit} = 14(0) + 6(13.33) = 0 + 79.98 = 79.98 \][/tex]
### Point B: [tex]\((30, 0)\)[/tex]
- Checking the constraints:
1. [tex]\( 30 \leq 40 - 3(0) \Rightarrow 30 \leq 40 \)[/tex] (True)
2. [tex]\( 30 \geq 12 - 0 \Rightarrow 30 \geq 12 \)[/tex] (True)
3. [tex]\( 30 \leq 30 - 0 \Rightarrow 30 \leq 30 \)[/tex] (True)
4. [tex]\( 30 \geq 0 \)[/tex] (True)
5. [tex]\( 0 \geq 0 \)[/tex] (True)
- All constraints are satisfied.
- Calculating the profit:
[tex]\[ \text{Profit} = 14(30) + 6(0) = 420 + 0 = 420 \][/tex]
### Point C: [tex]\((25, 5)\)[/tex]
- Checking the constraints:
1. [tex]\( 25 \leq 40 - 3(5) \Rightarrow 25 \leq 25 \)[/tex] (True)
2. [tex]\( 25 \geq 12 - 5 \Rightarrow 25 \geq 7 \)[/tex] (True)
3. [tex]\( 25 \leq 30 - 5 \Rightarrow 25 \leq 25 \)[/tex] (True)
4. [tex]\( 25 \geq 0 \)[/tex] (True)
5. [tex]\( 5 \geq 0 \)[/tex] (True)
- All constraints are satisfied.
- Calculating the profit:
[tex]\[ \text{Profit} = 14(25) + 6(5) = 350 + 30 = 380 \][/tex]
### Point D: [tex]\((12, 0)\)[/tex]
- Checking the constraints:
1. [tex]\( 12 \leq 40 - 3(0) \Rightarrow 12 \leq 40 \)[/tex] (True)
2. [tex]\( 12 \geq 12 - 0 \Rightarrow 12 \geq 12 \)[/tex] (True)
3. [tex]\( 12 \leq 30 - 0 \Rightarrow 12 \leq 30 \)[/tex] (True)
4. [tex]\( 12 \geq 0 \)[/tex] (True)
5. [tex]\( 0 \geq 0 \)[/tex] (True)
- All constraints are satisfied.
- Calculating the profit:
[tex]\[ \text{Profit} = 14(12) + 6(0) = 168 + 0 = 168 \][/tex]
### Conclusion
Comparing the profits from all points:
- Point A: [tex]\( 79.98 \)[/tex]
- Point B: [tex]\( 420 \)[/tex]
- Point C: [tex]\( 380 \)[/tex]
- Point D: [tex]\( 168 \)[/tex]
The point that maximizes John's profit is [tex]\((30, 0)\)[/tex] with a profit of [tex]\(420\)[/tex].
Thus, the point [tex]\((30, 0)\)[/tex] will maximize John's profit.
