High School

Lytbrosk West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting x apartments is represented by the following function:

\[ P(x) = 10x^2 + 1930x - 60000 \]

(a) What is the actual profit realized from renting the 41st unit, assuming that 40 units have already been rented? To calculate this, find \( P(41) - P(40) \).

(b) Compute the marginal profit when \( x = 40 \) and compare your results with that obtained in part (a).

Answer :

Final answer:

The actual profit realized from renting the 413th unit is $2,502,580. The marginal profit when x = 40 is $2,730.

Explanation:

To calculate the actual profit realized from renting the 413th unit, we substitute x = 413 into the profit function:

P(413) = 10(413)^2 + 1930(413) - 60000

Simplifying the expression:

P(413) = 10(170569) + 798890 - 60000

P(413) = 1705690 + 798890 - 60000

P(413) = 2502580

Therefore, the actual profit realized from renting the 413th unit is $2,502,580.

To compute the marginal profit when x = 40, we need to find the derivative of the profit function:

P'(x) = 20x + 1930

Substituting x = 40 into the derivative:

P'(40) = 20(40) + 1930

P'(40) = 800 + 1930

P'(40) = 2730

Therefore, the marginal profit when x = 40 is $2,730.

Comparing the results:

The actual profit from renting the 413th unit is $2,502,580, while the marginal profit when x = 40 is $2,730. The actual profit represents the total profit obtained from renting a specific unit, while the marginal profit represents the additional profit obtained from renting one more unit. The marginal profit is much smaller compared to the actual profit, indicating that the profit growth is slowing down as more units are rented.

Learn more about computing marginal profit here:

https://brainly.com/question/31254285

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