In a large bat factory, marginal revenues increase at first before decreasing rapidly as more bats are made. A limitation to keep in mind is the maximum capacity of the factory being 50 thousand bats per month. If the marginal revenue of making x thousand bats a month is 100-((18x)/((1+0.03x^2)^2)), find the revenue for making 10 thousand bats in a month. How do I enter this answer in in millions of dollars?
I understand that you find the antiderivative of the marginal revenue function but how do I calculate for C in order to get a function I can plug values into? Thats the part I get confused on.
You don't need to find C, since the total revenue is a definite integral.
∫[0,10] 100-((18x)/((1+0.03x^2)^2)) dx
= 100x + 30000/(3x^2+1000) [0,10] = 775
To find the revenue for making 10 thousand bats in a month, we need to calculate the total revenue. The total revenue can be found by multiplying the quantity (number of bats) by the marginal revenue.
Given the marginal revenue function:
MR(x) = 100 - (18x) / ((1 + 0.03x^2)^2)
To find the revenue for making 10 thousand bats (x = 10), we substitute x = 10 into the marginal revenue function:
MR(10) = 100 - (18 * 10) / ((1 + 0.03 * 10^2)^2)
Simplifying this equation:
MR(10) = 100 - (180) / ((1 + 0.03 * 100)^2)
= 100 - (180) / ((1 + 3)^2)
= 100 - (180) / (4^2)
= 100 - (180) / 16
= 100 - 11.25
= 88.75
Therefore, the marginal revenue for making 10 thousand bats in a month is 88.75.
To express this answer in millions of dollars, you need to divide it by 1 million:
88.75 divided by 1,000,000 = 0.00008875 million dollars
So, the revenue for making 10 thousand bats in a month is approximately 0.00008875 million dollars.