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.
R'(x)=-0.1x+40
(a) Find the daily total revenue realized from the sale of 180 units of the toaster oven.
$ ?
(b) Find the additional revenue realized when the production (and sales) level is increased from 180 to 280 units.
$ ?
(a) To find the daily total revenue realized from selling 180 units of the toaster oven, we need to integrate the marginal revenue function, R'(x), with respect to x, and evaluate it at x = 180.
Given the marginal revenue function:
R'(x) = -0.1x + 40
To find the total revenue function, we need to integrate R'(x) with respect to x:
R(x) = ∫(-0.1x + 40) dx
Integrating the function, we get:
R(x) = -0.1 * (x^2/2) + 40x + C
Where C is the constant of integration.
To find C, we can use the fact that R(0) = 0, since there would be no revenue if no units are sold:
0 = -0.1 * (0^2/2) + 40(0) + C
0 = C
Thus, the total revenue function becomes:
R(x) = -0.1 * (x^2/2) + 40x
Now we can find the daily total revenue realized from selling 180 units by substituting x = 180 into the total revenue function:
R(180) = -0.1 * (180^2/2) + 40(180)
R(180) = -0.1 * (16200/2) + 7200
R(180) = -0.1 * 8100 + 7200
R(180) = -810 + 7200
R(180) = 6390
Therefore, the daily total revenue realized from the sale of 180 units of the toaster oven is $6390.
(b) To find the additional revenue realized when the production level is increased from 180 to 280 units, we can calculate the difference between the total revenues of selling 280 units and 180 units.
Using the total revenue function we found earlier, R(x) = -0.1 * (x^2/2) + 40x, we can calculate the revenues for both cases:
R(180) = -0.1 * (180^2/2) + 40(180)
R(180) = 6390
R(280) = -0.1 * (280^2/2) + 40(280)
R(280) = 10780
The additional revenue realized when the production level is increased from 180 to 280 units is:
Additional Revenue = R(280) - R(180)
Additional Revenue = 10780 - 6390
Additional Revenue = 4390
Therefore, the additional revenue realized when the production level is increased from 180 to 280 units is $4390.