I'm having a bit of difficulty trying to solve this question for a part of my project. It involves the change in income in a year from a change in yield.

The first parts of the project have me optimizing the percent of land (out of 50 million acres) I should use for the max national income. I found that the percent I should use for soybeans (The segment of the related rates problem I am having problems with) was 13.29%. (So 6.645 E6 total acres used for growing soybeans)

Soybeans are harvested with a yield of 50 bushels an acre, initially.

The estimated selling price of a bushel of soybeans is modeled by equation:

p(S)=(3 E10)/((1 E9)+S)

Where S is the total amount of bushels of soybeans produced.

The cost of production for each bushel of soybeans is $10.10

The only problem I have now is with the related rates problem which asks me to find the annual change in income if the yield of soybeans is increasing at a rate of 1 bushel per acre every 3 years.

All in all:

6.645 E6 Acres used to grow soybeans.

50 bushels of soybeans per acre (Initially)

Yield of soybeans is increasing at a rate of 1 bushel every 3 years.

Profit from soybeans = ((3 E10)/(1 E9 +S)) - 10.1