Answer :
The marginal profit function is MP = -20x + 150, and for each additional apartment rented beyond 50 or 80, the firm will experience a decrease of $850 or $1450 in profit, respectively.
To find the marginal profit function, we need to subtract the cost function from the revenue function.
Revenue function: R = -10x² + 440x + 29000
Cost function: C = 290x + 1420
To find the profit function (P), we subtract the cost function from the revenue function:
P = R - C
= (-10x² + 440x + 29000) - (290x + 1420)
= -10x² + 440x + 29000 - 290x - 1420
= -10x² + 150x + 27580
Now, let's find the marginal profit function by taking the derivative of the profit function with respect to x:
MP = d(P)/dx = d(-10x² + 150x + 27580)/dx
= -20x + 150
The marginal profit function is MP = -20x + 150.
Now, let's calculate the marginal profit for different values of x:
1. When x = 50:
MP = -20(50) + 150
= -1000 + 150
= -850
The economic interpretation of this marginal profit at x = 50 is that for each additional apartment rented beyond 50, the firm will experience a decrease of $850 in profit.
2. When x = 80:
MP = -20(80) + 150
= -1600 + 150
= -1450
The economic interpretation of this marginal profit at x = 80 is that for each additional apartment rented beyond 80, the firm will experience a decrease of $1450 in profit.
In summary:
- The marginal profit function is MP = -20x + 150.
- At x = 50, the marginal profit is -$850.
- At x = 80, the marginal profit is -$1450.
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