3.4 Further Application of Optimization 1. ECONOMICS: Oil Prices: An oil-produci

Question

3.4 Further Application of Optimization

1. ECONOMICS: Oil Prices: An oil-producing country can sell 7 million barrels of oil a day at a price of $90 per barrel. If each $1 price increase will result in a sales decrease of 100,000 barrels per day, what price will maximize the country’s revenue? How many barrels will it sell at that price?

2. ENVIRONMENTAL SCIENCES: Maximum Yield: A peach grower finds that if he plants 40 trees per acre, each tree will yield 60 bushels of peaches. He also estimates that for each additional tree that he plants per acre, the yield of each tree will decrease by 2 bushels. How many trees should he plant per acre to maximize his harvest?

3. ENVIRONMENTAL SCIENCES: Maximum Yield: An apple grower finds that if she plants 20 trees per acre, each tree will yield 90 bushels of apples. She also estimates that for each additional tree that she plants per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

Explanation / Answer

revenue=price×quantity

The price is 90+x, where x is the change in price, and the quantity is 7,000,000100,000x (I'm going to assume that's what you meant when you wrote (7100,000x)). Now just take the derivative of that function.

r(x)=(90+x)(7,000,000100,000x)=100,000x22,000,000x+630,000,000r(x)=200,000x2,000,000

. For r(x) to be at either a maximum or a minimum at a given x, r(x) must be 0 at that point, so set r(x)=0 and see where this occurs (we'll call that point x1). This doesn't guarantee that r(x) is maximized at x1, just that it can be, so you need to check to make sure it is. If r(x)>0 before x1 and <0 after, you know r(x) is increasing up to x1 and decreasing after, which means it is in fact a local maximum. To find the global maximum, see if there is more than one x for which r(x) is locally maximized, and compare the values of r(x) at all those points. The x for which r(x) is highest of those points will be your global maximum, and that's where revenue is maximized. The revenue-maximizing price, therefore, is 90+x, and the number of barrels sold is 7,000,000100,000x.

Related Questions

Navigate

• Browse (All)
• Browse 3
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.