Calculus and Optimization Practice Problems 1,000 5,000 P( x) = + 100 1. Suppose a company makes a profit of X X makes and sells x > 0 items. How many items should it make to maximize profit? P( x) = 1,000 5,000 + 100 2. Suppose the profit of a company is when it makes X items a day. What level of production will maximize profits? 3. When 30 orange trees are planted on an acre, each will produce 500 oranges a year. For every additional orange tree planted, each tree will produce 10 fewer oranges. How many trees should be planted to maximize the yield? 4. An artist can sell 20 copies of a painting at $100 each, but for each additional copy she makes, the value of each painting will go down by a dollar. Thus, if 22 copies are made, each will sell for $98. How many copies should she make to maximize her sales? 5. A garden has 200 pounds of watermelons growing in it. Every day, the total amount of watermelon increases by 5 pounds. At the same time, the price per pound of watermelon goes down by 1c. If the current price is 90c per pound, how much longer should the watermelons grow in order to fetch the highest price possible? 6. A farmer has 400 feet of fencing to make three rectangular pens. What dimensions x and y will maximize the total area? P( x) = - 1,000 5,000 + 100 X x 7. Four pens will be built along a river by using 150 feet of fencing. What dimensions will maximize the area of the pens? P( x) =- 1,000 5,000 + 100 X 8. A rectangular pen will be built using 100 feet of fencing. What dimensions will maximize the area