To determine the maximum number of days each developer can work within the budget, we need to solve a linear optimization problem. We have three types of developers, each with a different daily rate, and we are given the constraints:

• Sr Dev: $224/day
• TA: $280/day (with a maximum of 3 days)
• Dev: $192/day

We need to find the maximum number of days each developer can work without exceeding the total monthly budget of $7160. Additionally, no developer can work more than 20 days.

Let's define:

• $x_1$ = number of days the Sr Dev works
• $x_2$ = number of days the TA works (with a maximum of 3 days)
• $x_3$ = number of days the Dev works

Objective:

Maximize the total number of working days: $x_1 + x_2 + x_3$

Constraints:

1. Budget constraint: $224x_1 + 280x_2 + 192x_3 \leq 7160$
2. Maximum working days for each developer: $x_1 \leq 20$, $x_2 \leq 3$, $x_3 \leq 20$
3. Non-negativity: $x_1 \geq 0$, $x_2 \geq 0$, $x_3 \geq 0$

We can solve this by testing the maximum allowable days within these constraints.

Let's proceed step by step:

1. Set $x_2$ to 3 (the maximum allowable days for TA) because TA has the highest rate and is limited.
2. Maximize $x_1$ and $x_3$ within the remaining budget.

Let's calculate:

• For $x_2 = 3$:

  • TA cost: $280 \times 3 = 840$
  • Remaining budget: $7160 - 840 = 6320$

• Maximize $x_1 + x_3$ under the new budget constraint:

  • $224x_1 + 192x_3 \leq 6320$

Now, we check different combinations for $x_1$ and $x_3$:

If $x_1 = 20$, then:

• Cost of Sr Dev: $224 \times 20 = 4480$
• Remaining budget: $6320 - 4480 = 1840$
• Maximum $x_3$ given remaining budget: $\frac{1840}{192} \approx 9.58$ (rounded down to 9 days)

Let's verify:

• Total days:
  • $x_1 = 20$, $x_2 = 3$, $x_3 = 9$
• Total cost:
  • $224 \times 20 + 280 \times 3 + 192 \times 9 = 4480 + 840 + 1728 = 7048$

This fits within the budget, and it's the maximum combination.

Solution:

• Sr Dev: 20 days
• TA: 3 days
• Dev: 9 days

This allocation will utilize $7048 of the $7160 budget.

Would you like any further details or have any questions?

Relative Questions:

1. How would the solution change if the budget increased?
2. What if TAs could work more than 3 days?
3. How would changing the daily rate of one of the developers impact the solution?
4. What happens if we require that each developer works at least 5 days?
5. How can we optimize the days if fractional days were allowed?

Tip:

When dealing with linear optimization problems, identifying constraints and systematically testing possible combinations can lead to an optimal solution.