A company charges $200 for each leather bag on order of 150 or less. The cost of each bag is reduced by $1 for each order in excess of 150 pieces. How many bags on order would result in maximum revenue? What is the maximum revenue?
the price, for n >= 150, is
p(n) = 200-n
Revenue is price*quantity, so
r(n) = n(200-n) = 200n-n^2
That's just a parabola. The vertex is the point of maximum revenue. That's just algebra I, so no sweat.
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right?
To find the number of bags on order that would result in maximum revenue, we need to understand the relation between the revenue and the number of bags.
Let's break down the problem:
1. For each order of 150 or less bags, the company charges $200 per bag. So, the revenue in this case would be 150 multiplied by $200, which is $30,000.
2. For orders exceeding 150 bags, the company reduces the cost of each bag by $1 for each additional bag. This means that for every extra bag beyond 150, the cost of each bag reduces by $1.
We can set up a mathematical equation to represent the revenue (R) in terms of the number of bags on order (N):
R(N) = (200 - (N-150)) * N
Now, let's find the number of bags on order that would result in maximum revenue:
To find the maximum revenue, we can differentiate the revenue equation (R) with respect to the number of bags on order (N) and set it equal to zero (dR/dN = 0). This will give us the critical point where the revenue is maximized.
dR/dN = 200 - 2N = 0
Solving this equation, we find:
200 - 2N = 0
2N = 200
N = 100
So, the critical point where revenue is maximized is when there are 100 bags on order.
To ensure that this is indeed a maximum, we can check the second derivative of the revenue equation by calculating:
d^2R/dN^2 = -2
Since the second derivative is negative, we can conclude that the revenue is maximized at N = 100.
Now, let's find the maximum revenue:
Substituting N = 100 into the revenue equation:
R(N) = (200 - (100-150)) * 100
R(N) = (200 - (-50)) * 100
R(N) = 250 * 100
R(N) = $25,000
Hence, the number of bags on order that would result in the maximum revenue is 100 bags, and the maximum revenue would be $25,000.