A manufacturer sells a certain article to dealers at a rate of $20 each if less than 50 are ordered. If 50 or more are ordered(up to 600) the price per article is reduced at a rate of 2 cents times the number ordered. What size order will produce the maximum amount of money for the manufacturer?
To determine the order size that will produce the maximum amount of money for the manufacturer, we need to find the point where the revenue is maximized.
Let's start by setting up the revenue function. The revenue is calculated by multiplying the price per article by the number of articles sold.
For orders less than 50, the price per article is $20. Therefore, the revenue function when the order is less than 50 can be defined as:
R(x) = 20x
For orders of 50 or more, the price per article is reduced by 2 cents for each article ordered. The revenue function when the order is 50 or more can be defined as:
R(x) = (20 - 0.02x)x
Combining the two cases, we can define the revenue function as follows:
R(x) = 20x if x < 50
R(x) = (20 - 0.02x)x if x >= 50
Now, we can find the order size that maximizes the revenue. To do that, we need to find the derivative of the revenue function and set it equal to zero:
For the first case when x < 50:
R'(x) = 20
For the second case when x >= 50:
R'(x) = (20 - 0.04x)
Setting R'(x) equal to zero for the second case:
(20 - 0.04x) = 0
0.04x = 20
x = 500
Therefore, the order size that maximizes the revenue for the manufacturer is 500.
To find the order size that will produce the maximum amount of money for the manufacturer, we need to consider the pricing structure and the relationship between the number of articles ordered and the total revenue.
First, let's determine the cost per article depending on the order size:
- For orders of less than 50 articles, the cost per article is fixed at $20.
- For orders of 50 or more articles, the cost per article reduces by 2 cents times the number ordered.
Now, let's analyze the total revenue generated by each order size:
- For orders of less than 50 articles, the total revenue is calculated by multiplying the cost per article ($20) by the order size.
- For orders of 50 or more articles, the total revenue is calculated by multiplying the reduced cost per article ($20 - 0.02 * order size) by the order size.
To find the order size that maximizes the manufacturer's revenue, we can create an equation for the total revenue as a function of the order size:
For orders of less than 50 articles: Revenue = 20 * OrderSize
For orders of 50 or more articles: Revenue = (20 - 0.02 * OrderSize) * OrderSize
To find the maximum revenue, we can differentiate the revenue function with respect to the order size and set it equal to zero:
dRevenue/dOrderSize = 0
Taking the derivative of the revenue function gives us:
For orders of less than 50 articles: dRevenue/dOrderSize = 20
For orders of 50 or more articles: dRevenue/dOrderSize = 20 - 0.04 * OrderSize
Now we can solve for the order size that maximizes the revenue by setting the derivative equal to zero:
For orders of 50 or more articles: 20 - 0.04 * OrderSize = 0
Solving this equation gives us:
0.04 * OrderSize = 20
OrderSize = 20 / 0.04
OrderSize = 500
Therefore, an order size of 500 articles will produce the maximum amount of money for the manufacturer.
I think I am reading this wrong but this may not be a Calculus question.
<= means less than or equal to
>= means greater than or equal to
Rates
20*x 0 < x < 50
20-0.02*x 50 <= x <= 600
When buying 1-49 articles, it is $20 per article.
When buying 50-600 articles, it is $(20-.02*x).
As you buy more, the cheaper it gets until you reach 600, then it maintains at that cost which is $8 dollars per article.