The correct statements are:
- The country can increase its output of beef from 0 tons to 30 tons by sacrificing 30 tons of vegetables.
- The slope of the production possibilities curve is 1.
If a production possibilities curve shows that the United States has an opportunity-cost ratio of 1 ton of vegetables for 1 ton of beef, it means that the country can produce 1 ton of beef by sacrificing 1 ton of vegetables, and vice versa. This is reflected in the slope of the production possibilities curve, which is constant and equal to -1 (or 1, if we take the absolute value).
Additionally, if the country is currently producing 0 tons of beef, it could increase its output to 30 tons of beef by sacrificing 30 tons of vegetables, since the opportunity cost of producing 1 ton of beef is 1 ton of vegetables.
The statement "the slope of the production possibilities curve is 30" is not correct, as the slope is always equal to the opportunity-cost ratio, which is 1 in this case.
The statement "the country is unlikely to make a profit if it produces an equal amount of beef and vegetables" is not necessarily correct, as it depends on the market demand and prices for beef and vegetables. The production possibilities curve only shows the maximum combination of beef and vegetables that can be produced with the given resources and technology.