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) Total revenue realized from the sale of 180 units of the toaster oven: $-72.00
(b) Additional revenue realized when the production (and sales) level is increased from 180 to 280 units: $-80.00
To find the daily total revenue realized from the sale of 180 units of the toaster oven, you need to integrate the daily marginal revenue function.
Given: R'(x) = -0.1x - 40
To find the total revenue, you need to integrate R'(x) with respect to x. The integral of -0.1x is (-0.1x^2)/2 = -0.05x^2/2 = -0.025x^2. And the integral of -40 is -40x.
So, the total revenue function R(x) = -0.025x^2 - 40x.
(a) To find the daily total revenue from selling 180 units, substitute x = 180 into the total revenue function.
R(180) = -0.025(180)^2 - 40(180)
= -0.025(32400) - 7200
= -810 + 7200
= $6390
Therefore, the daily total revenue realized from the sale of 180 units is $6390.
(b) To find the additional revenue when the production level is increased from 180 to 280 units, find the difference between the total revenue at x = 280 and x = 180.
Additional revenue = R(280) - R(180)
= [-0.025(280)^2 - 40(280)] - [-0.025(180)^2 - 40(180)]
= [-1960 - 11200] - [-810 - 7200]
= -13160 + 8010
= $5140
Therefore, the additional revenue realized when the production level is increased from 180 to 280 units is $5140.
To find the daily total revenue from the sale of 180 units of the toaster oven, we need to integrate the marginal revenue function.
Let's start with part (a):
(a) Find the daily total revenue realized from the sale of 180 units of the toaster oven.
To find the total revenue, we need to integrate the marginal revenue function R '(x).
R '(x) = -0.1x - 40
To find the total revenue, we integrate R'(x) with respect to x:
R(x) = ∫(-0.1x - 40) dx
Integrating, we get:
R(x) = -0.1 * (x^2 / 2) - 40x + C
Where C is the constant of integration. Since we are given the production level is 180 units, we can use this information to find C.
When x = 180, R(x) is the daily revenue. So we have:
R(180) = -0.1 * (180^2 / 2) - 40 * 180 + C
Now, we can calculate this expression to find the daily total revenue:
R(180) = -0.1 * (16200 / 2) - 7200 + C
= -0.1 * 8100 - 7200 + C
= -810 - 7200 + C
= -8010 + C
Therefore, the daily total revenue realized from the sale of 180 units of the toaster oven is -$8010 + C dollars.
For part (b):
(b) Find the additional revenue realized when the production level is increased from 180 to 280 units.
To find the additional revenue, we need to compare the total revenue at the production levels of 280 and 180 units.
Let's calculate the total revenue at x = 280 units:
R(280) = -0.1 * (280^2 / 2) - 40 * 280 + C
= -0.1 * 39200 - 11200 + C
= -3920 - 11200 + C
= -15120 + C
And now, let's calculate the total revenue at x = 180 units:
R(180) = -0.1 * (180^2 / 2) - 40 * 180 + C
= -0.1 * 16200 - 7200 + C
= -1620 - 7200 + C
= -8820 + C
Therefore, the additional revenue realized when the production level is increased from 180 to 280 units is:
R(280) - R(180) = (-15120 + C) - (-8820 + C)
= -15120 + C + 8820 - C
= -15120 + 8820
= -$6300
Therefore, the additional revenue realized when the production level is increased from 180 to 280 units is -$6300 dollars.