To solve this problem, we need to set up a system of inequalities and an objective function to maximize the daily profit of the skateboard factory.

Step 1: Define Variables

• Let $M$ represent the number of Mega skateboards produced daily.
• Let $S$ represent the number of Supreme skateboards produced daily.

Step 2: Write the Constraints

1. The total number of skateboards produced cannot exceed 40: $M + S \leq 40$

2. The factory must produce at least 3 times as many Mega skateboards as Supreme skateboards: $M \geq 3S$

3. Both $M$ and $S$ must be non-negative: $M \geq 0 \quad \text{and} \quad S \geq 0$

Step 3: Define the Objective Function

• The profit from Mega skateboards is $55 per board, so the profit from $M$ Mega skateboards is $55M$.
• The profit from Supreme skateboards is $75 per board, so the profit from $S$ Supreme skateboards is $75S$.

The objective function (total profit $P$) to maximize is: $P = 55M + 75S$

Step 4: Solve the System of Inequalities We need to find the values of $M$ and $S$ that maximize $P = 55M + 75S$ under the constraints.

Step 5: Check the Vertices of the Feasible Region The feasible region is determined by the constraints:

1. $M + S = 40$
2. $M = 3S$

Now, let's find the intersection points:

1. Set $M = 3S$ into $M + S = 40$: $3S + S = 40 \implies 4S = 40 \implies S = 10$ Substituting $S = 10$ into $M = 3S$: $M = 3 \times 10 = 30$ So, one vertex is $(M, S) = (30, 10)$.

2. Other vertices are found by testing when $S = 0$:

   • If $S = 0$, then $M + 0 = 40$ implies $M = 40$.
   • If $M = 0$, then $S = 0$ because $M = 3S$.

Step 6: Evaluate the Profit at Each Vertex

1. At $(M, S) = (30, 10)$: $P = 55 \times 30 + 75 \times 10 = 1650 + 750 = 2400$

2. At $(M, S) = (40, 0)$: $P = 55 \times 40 + 75 \times 0 = 2200$

3. At $(M, S) = (0, 0)$: $P = 55 \times 0 + 75 \times 0 = 0$

Step 7: Conclusion The maximum profit occurs at $(M, S) = (30, 10)$, with a maximum daily profit of:

$\boxed{2400}$

Would you like more details on any part of this solution, or do you have any questions?

Here are some related questions for further exploration:

1. How would the maximum profit change if the profit per Mega skateboard increased to $60?
2. What would happen to the maximum profit if the constraint $M \geq 3S$ changed to $M \geq 2S$?
3. How would