Respuesta :
Answer:
66
Step-by-step explanation:
Certainly! Let's find the marginal revenue and interpret the result in the context of this scenario.
Finding Marginal Revenue:
Marginal revenue refers to the additional revenue earned by selling one more unit of a product. It's essentially the derivative of the revenue function R(x).
In this case, the revenue function is given as:
R(x) = (58x^2 + 70x) / (2x + 2)
To find the marginal revenue, we need to differentiate R(x) with respect to x. Due to the rational expression, we can use quotient rule differentiation.
However, there's a shortcut we can employ here. Notice that the denominator (2x + 2) is also present in the numerator. Dividing both the numerator and denominator by (2x + 2) simplifies the expression significantly, and the quotient rule differentiation becomes unnecessary.
Simplified Revenue Function:
R(x) = 58x + 70
Differentiating for Marginal Revenue:
Since this is a simple linear function, the derivative (marginal revenue) is simply the coefficient of the x term:
MR(x) = dR(x)/dx = 58
Marginal Revenue When 65 Units Sold:
We are interested in the marginal revenue when 65 units are sold (x = 65). Plugging this value into the marginal revenue function:
MR(65) = 58
Interpretation:
The marginal revenue, MR(65) = $58, signifies that when 65 units have already been sold, selling one more unit (the 66th unit) would yield an additional revenue of approximately $58. In other words, the company's total revenue would increase by $58 if they were able to sell one more unit at the current price point after selling 65 units.
Projected Revenue from Selling Unit 66:
While the marginal revenue tells us the additional revenue from selling one more unit, it doesn't directly give the projected total revenue from selling unit 66.
To find the projected revenue from selling unit 66, we need to calculate the total revenue when 66 units are sold (R(66)) and subtract the total revenue when 65 units are sold (R(65)).
However, due to the complexity of the original revenue function (before simplification), calculating R(66) and R(65) might be cumbersome.
Alternative Approach:
We can leverage the concept of marginal revenue here. Since the marginal revenue represents the additional revenue from selling one more unit, we can add the marginal revenue we just found (MR(65) = $58) to the total revenue when 65 units are sold (R(65)).
This approach assumes the marginal revenue remains relatively constant around the point of 65 units sold, which might be a reasonable assumption if the function doesn't have sharp changes in its slope.
Therefore, the projected total revenue from selling unit 66 would be approximately:
Projected Revenue (Unit 66) ≈ R(65) + MR(65) = $58 (marginal revenue) + R(65) (unknown total revenue at 65 units)
Note: Without the actual expression for R(x), we cannot calculate the exact total revenue at 65 units (R(65)). However, the approach above provides an approximate estimate for the projected revenue from selling unit 66.
