The management of Ditton Industries has determined that the daily marginal revenue function associated with selling x units of their deluxe toaster ovens is given by the following where R '(x) is measured in dollars/unit.
Rtext( )'\(x\) = -0.1x + 40
(a) Find the daily total revenue realized from the sale of 182 units of the toaster oven.
To find the daily total revenue realized from the sale of 182 units of the toaster oven, we need to integrate the marginal revenue function with respect to x over the interval from 0 to 182.
The marginal revenue function is given by:
R'(x) = -0.1x + 40
To integrate this function, we first need to find its antiderivative. Integrating -0.1x + 40 with respect to x, we get:
R(x) = -0.1 * (x^2/2) + 40x + C
The constant of integration, C, is a constant term that appears when integrating a function. However, since we only need to find the daily total revenue for a specific number of units sold, we don't need to worry about the constant of integration in this case.
Now, we can substitute the value of x = 182 into the total revenue function R(x):
R(182) = -0.1 * (182^2/2) + 40 * 182
Simplifying this expression, we get the daily total revenue realized from the sale of 182 units of the toaster oven.