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 183 units of the toaster oven.
(b) Find the additional revenue realized when the production (and sales) level is increased from 183 to 283 units.
R(x) = ∫R'(x) dx = -0.05x^2 + 40x
Now just plug in your x values.
(a) To find the daily total revenue realized from the sale of 183 units, we first need to find the total revenue function. The marginal revenue function R'(x) is the derivative of the total revenue function R(x).
To find the total revenue function R(x), we integrate R'(x) with respect to x:
R(x) = ∫[-0.1x + 40] dx
Integrating each term separately:
R(x) = ∫-0.1x dx + ∫40 dx
Using the power rule of integration:
R(x) = -0.1 ∫x dx + 40x + C
Integrating x:
R(x) = -0.1 * (x^2/2) + 40x + C
Simplifying:
R(x) = -0.05x^2 + 40x + C
Now, we can find the daily total revenue realized from selling 183 units by substituting x = 183 into the total revenue function:
R(183) = -0.05(183)^2 + 40(183) + C
We don't have the value of C, but it is not required to compute the value of R(183).
R(183) = -0.05(33,489) + 7,320 + C
R(183) = -1,674.45 + 7,320 + C
R(183) = 5,645.55 + C
Therefore, the daily total revenue realized from selling 183 units of the toaster oven is $5,645.55 + C.
(b) To find the additional revenue realized when the production (and sales) level is increased from 183 to 283 units, we need to calculate the difference in total revenue between these two production levels.
First, we find the total revenue function R(x) by integrating R'(x):
R(x) = -0.05x^2 + 40x + C
To find the additional revenue, we subtract the total revenue at x = 183 from the total revenue at x = 283:
Additional revenue = R(283) - R(183)
Additional revenue = [-0.05(283)^2 + 40(283) + C] - [-0.05(183)^2 + 40(183) + C]
Additional revenue = [-0.05(79,489) + 11,320 + C] - [-0.05(33,489) + 7,320 + C]
Additional revenue = -3,974.45 + 11,320 + C - (-1,674.45 + 7,320 + C)
Additional revenue = 7,345.55
Therefore, the additional revenue realized when the production (and sales) level is increased from 183 to 283 units is $7,345.55.
To find the daily total revenue realized from the sale of 183 units of the toaster oven, we need to integrate the marginal revenue function R'(x).
(a) First, let's integrate the marginal revenue function R'(x) = -0.1x + 40 to find the total revenue function R(x):
R(x) = ∫(R'(x)) dx
R(x) = ∫(-0.1x + 40) dx
The integral of -0.1x is -0.05x^2, and the integral of 40 is 40x. Adding these terms together, we have:
R(x) = -0.05x^2 + 40x
Now we can substitute x = 183 into the total revenue function to find the daily total revenue:
R(183) = -0.05(183)^2 + 40(183)
R(183) = -0.05(33489) + 7320
R(183) = -1674.45 + 7320
R(183) = 5645.55
Therefore, the daily total revenue realized from the sale of 183 units of the toaster oven is $5,645.55.
(b) To find the additional revenue realized when the production (and sales) level is increased from 183 to 283 units, we need to find the difference in total revenue between these two production levels.
Let's first find the total revenue for 183 units:
R(183) = -0.05(183)^2 + 40(183)
R(183) = -0.05(33489) + 7320
R(183) = -1674.45 + 7320
R(183) = 5645.55
Now let's find the total revenue for 283 units:
R(283) = -0.05(283)^2 + 40(283)
R(283) = -0.05(80089) + 11320
R(283) = -4004.45 + 11320
R(283) = 7315.55
The additional revenue realized when the production level is increased from 183 to 283 units is:
Additional Revenue = R(283) - R(183)
Additional Revenue = 7315.55 - 5645.55
Additional Revenue = 1670
Therefore, the additional revenue realized when the production level is increased from 183 to 283 units is $1670.