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Name ______________________________________________Date __________________________________Thurgood Marshall Academy Public Charter High School – CalculusNotes #37 – More Optimization1. On a certain route, an airline carries 8000 passengers per month, each paying $50. The airline wants to increase the fare. However, the market research department estimates that for each $1 increase in fare, the airline will lose 100 passengers. Determine the price which maximizes the airline’s revenue.2. A swimming club offers memberships at the rate of $200, provided that a minimum of 100 people join. For each member in excess of 100, the membership fee will be reduced by $1 per person (for each member). At most, 160 memberships will be sold. How many memberships should the club try to sell in order to maximize its revenue?3. In the planning of a sidewalk café, it is estimated that if there are 12 tables, the daily profit will be $10 per table. Because of overcrowding, for each additional table the profit per table (for every table in the café) will be reduced by $.50. How many tables should be provided to maximize the profit from the café?4. A certain toll road averages 36,000 cars per day when charging $1 per car. A survey concludes that increasing the toll will result in 300 fewer cars for each cent of increase. What toll should be charged in order to maximize the revenue?5. The manager of a department store wants to build a 600-square-foot rectangular enclosure on the store’s parking lot in order to display some equipment. Three sides of the enclosure will be built of redwood fencing, at a cost of $7 per foot. The fourth side will be built of cement blocks, at a cost of $14 per foot. Find the dimensions of the enclosure that will minimize the total cost of the building materials.6. Three hundred and twenty dollars is available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $6 per foot and the fencing for the other three sides costs $2 per foot. Find the dimensions of the garden with the largest possible area.7. A rectangular garden of area 75 square feet is to be surrounded on three sides by a brick wall costing $10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden such that the cost of materials is minimized.8. Starting with a 100-foot-long stone wall, a farmer would like to construct a rectangular enclosure by adding 400 feet of fencing. Find the dimensions of the rectangle that result in the greatest possible area.9. An open-top rectangular box must have a square base. The volume of the box must be 32 cubic feet. Find the dimensions of the box so that the total surface area will be as small as possible.10. A closed rectangular box with square base and a volume of 12 cubic feet is to be constructed using two different types of materials. The top is made of a metal costing $2 per square foot and the remainder of wood costing $1 per square foot. Find the dimensions of the box for which the cost of materials is minimized.11. An open-top rectangular box is to be constructed with a surface area of 100 square cm. Find the dimensions of the box so that the volume is as large as possible. ................
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