Question

The marginal revenue for a new calculator is given by

MR = 90,000 − 30,000 / (10 + x)²
where x represents hundreds of calculators and revenue is in dollars. Find the total revenue function, R(x), for these calculators.

Answer 1

The total revenue function R(x) is obtained by integrating the marginal revenue function MR = 90,000 - 30,000 / (10 + x)². Through integration, the total revenue function R(x) is determined up to an integration constant, which can be set with initial revenue conditions.

Step-by-step explanation:

The question revolves around finding the total revenue function, R(x), from the given marginal revenue equation, MR = 90,000 - 30,000 / (10 + x)², where x represents hundreds of calculators. Since marginal revenue is the derivative of total revenue, to find the total revenue function, we need to integrate the marginal revenue function with respect to x. The integration will provide us the total revenue function up to a constant C, which can be determined if additional information, like initial revenue, is provided.

To integrate MR with respect to x, we simplify the expression and integrate:

1. Firstly, rewrite the MR function: MR = 90,000 - 30,000 / (100 + 10x)²
2. Next, integrate with respect to x:
3. R(x) = ∫ MR dx = ∫ (90,000 - 30,000 / (100 + 10x)²) dx
4. Integration leads to a general form of the total revenue function, R(x) = 90,000x - (30,000 / (10 * (10 + x))) + C

The student can use this function to evaluate total revenue for any quantity x of hundreds of calculators sold, bearing in mind that the constant C would typically be determined by initial conditions such as total revenue when x is zero.