Find the marginal profit (dollars) for selling x units

Find the marginal profit (dollars) for selling x units P = 0.43x^3 + 10x^2 – 3x – 2950 20x^2 + 1.29x – 3 1.29x^2 + 20x – 3 Submit Question Question 13 pt 0 Details Given P = 0.43x^3 + 10x^2 – 3x – 2950 What is the marginal profit for selling 210 units?
50,086
61,086

The Correct Answer and Explanation is:

Correct Answer for the first question: 1.29x^2 + 20x – 3
Correct Answer for Question 13: $61,086

Explanation

The problems presented require finding a marginal profit function and then using it to calculate the marginal profit at a specific production level. In economics and business calculus, the marginal profit is the derivative of the total profit function. It represents the rate of change of profit with respect to the number of units sold. Essentially, it provides an excellent approximation of the additional profit that would be generated by selling one more unit.

Part 1: Finding the Marginal Profit Function

The given total profit function, P, for selling x units is:
P(x) = 0.43x³ + 10x² – 3x – 2950

To find the marginal profit function, which we will denote as P'(x), we must take the first derivative of the profit function P(x) with respect to x. We can do this by applying the power rule of differentiation to each term of the polynomial. The power rule states that the derivative of a term axⁿ is n·a·xⁿ⁻¹.

Let’s differentiate the profit function term by term:

1. The derivative of 0.43x³ is (3) * (0.43) * x^(3-1) = 1.29x².
2. The derivative of 10x² is (2) * (10) * x^(2-1) = 20x.
3. The derivative of -3x is (1) * (-3) * x^(1-1) = -3x⁰ = -3 (since any non-zero number raised to the power of 0 is 1).
4. The derivative of the constant term -2950 is 0, as the rate of change of a constant is always zero.

Combining these results, we get the marginal profit function:
P'(x) = 1.29x² + 20x – 3

This matches the fourth option provided in the first part of the quiz.

Part 2: Calculating the Marginal Profit for Selling 210 Units

The second question asks for the specific value of the marginal profit when 210 units are sold. To find this, we substitute x = 210 into the marginal profit function P'(x) that we just derived.

P'(x) = 1.29x² + 20x – 3
P'(210) = 1.29(210)² + 20(210) – 3

Now, we perform the calculation:

1. First, calculate (210)²: 210 * 210 = 44,100.
2. Next, multiply this result by 1.29: 1.29 * 44,100 = 56,889.
3. Then, calculate the middle term: 20 * 210 = 4,200.
4. Finally, substitute these values back into the equation and perform the addition and subtraction:
   P'(210) = 56,889 + 4,200 – 3
   P'(210) = 61,089 – 3
   P'(210) = 61,086

The marginal profit for selling 210 units is $61,086. This figure indicates that selling the 211th unit is expected to increase the total profit by approximately $61,086. This corresponds to the fourth option in Question 13.