5. Find two numbers whose sum is 48 and whose product is to be a maximum.

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2 Optimization Practice (4.4) 1. If 40 passengers hire a special car on a train, they will be charged $8 each. This fare will be reduced by $.10 each passenger, for each person in addition to these 40. What number of passengers will produce the maximum profit for the railroad?. A fruit grower estimates that if he harvests his crop of oranges now, he will get 100 pounds per tree, which he can sell for $.5 per pound. For each week he waits, he estimates that the crop will increase by 10 lb. per tree, but the price will decrease by $.01 per week. When should he pick the oranges to obtain the maximum profit? What would his profit be at this time? 3. A rectangular box with a square base and a cover is to be built to contain 640 cubic feet. If the cost per square foot for the bottom is $15 and for the top and the sides is $10, what is the minimum cost of the constructed box? 4. A tinsmith wishes to make an open box from a square piece of tin which measures 8 by 8. To accomplish this task, he proposes to cut equal square pieces from each corner of the tin and fold up the tin to form sides. Determine the sides of the squares to be cut from the corners so that the box will have the greatest possible volume. What is this volume? 5. Find two numbers whose sum is 48 and whose product is to be a maximum. 6. Suppose that a rancher has 1000 feet of fencing available to make a rectangular corral. A barn will form one side of the corral so no fencing will be needed there. What dimensions will give the maximum area? 7. Allen Rent-A-TV derives an average profit of $15 per customer if it services 1000 customers or less. If it services over 1000 customers, the profit decrease per customer by $.01 for each customer over How many customers will give the maximum profit?

3 3 8. The Dobbs Hotel will provide a dinner party for a minimum of 100 couples at $50 per couple. If more than 100 couples attend, the hotel will refund every couple $.5 for every couple over 100. How many couples will maximize the hotel s revenue? 9. A poster is to contain 50 square inches of printed matter with margins of 4 each at the top and bottom and at each side. Find the overall dimensions if you want a minimum total area. 10. An oil can is to be made in the form of a right circular cylinder to contain 16π cubic inches. What dimensions of the can will require the least amount of material, while meeting this requirement for volume. (Recall that for a cylinder, V = π r h, and area of a circle A= π r.) 11. The yield of orange trees is reduced if they are planted too close together. If there are 30 trees per acre, each tree produces 400 oranges. For each additional tree in the acre, the yield is reduced by 7 oranges per tree. How many trees per acre yield the largest crop for farmer Boyles? 1. A rectangular box is to be made from a piece of cardboard 4 inches long and 9 inches wide by cutting out identical squares from the four corners and turning up the cardboard to form the sides. What size square should you cut off of each corner to maximize the volume of the box? What is this maximum volume? 13. Rancher Sellers has 80 ft. of fence with which he plans to enclose a rectangular pen along one side of his 100 ft. barn (the side along the barn needs no fence). What are the dimensions that would maximize the area? What is the maximum area? 14. A handbill is to contain 50 square inches, with 4 inch margins at the top and bottom and inch margins on each side. What dimensions for the handbill would give the largest printed area? 15. A man with 300 m of fencing wishes to enclose a rectangular area and divide it into 5 pens with fences parallel to one side. (See figure at right) What dimensions would maximize the area?

4 4 Related Rates Class Notes, Day 1 - AB Calculus Keys to related rates: 1) Set up an equation that relates all your. 1a) You may have to set up two equations and one into the other. 1b) You can fill in anything you KNOW remains over time. ) Take the derivative with respect to, because things are changing over. 3) Fill in the things you know and solve for the. Typical example #1 TRIANGLES (use ) A 6 m. ladder is against the wall. If its bottom is pushed/pulled at a constant 1 m/sec, how fast is the ladder top sliding when it reaches 5 m. up the wall? 3 m.? A winch (altitude 0 ft) reels in a rope at ft/sec. How fast is the boat moving when the rope is 45 feet long? When the rope is 30 feet? Typical example # SIMILAR TRIANGLES (use ) A man 6 feet tall walks away from a lamp post 15 feet tall at a rate of 5 ft/sec. How fast is his shadow lengthening? How fast is the shadow s tip moving?

5 5 Typical example #3 CONE PROBLEMS (use a sub-relationship) Water is flowing into a cone (height = 16 cm and radius = 4 cm) at a rate of cubic cm per minute. How fast is the water level rising when the water is 5 cm deep? 10 cm deep? Typical example #4 ANGLES (use,, or ) A plane is flying at an altitude of 4000 ft and is flying west at a rate of 700 ft/sec. A searchlight, under its path, tracks it. How fast is the light pivoting when the plane is 1000 ft east? Overhead? 4000 ft Example #5 The product of positive # s is 30. The sum of the first and 5 times the second is minimized. Find the # s. Example #6 Calculate the dimensions of the rectangle with maximum area that can be inscribed in a circle of diameter 6 cm.

6 Related Rates DAY 1 6 1) The top of a 5-foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of 1 foot per minute. How fast is the bottom of the ladder slipping along the ground when the bottom of the ladder is 7 feet away from the base of the wall? ) A balloon is being inflated by pumping air in at the rate of cubic inches per second. How fast is the diameter of the balloon increasing when the radius is one-half inch? 3) Oil from an uncapped oil well in the ocean is radiating outward in the form of a circular film on the surface of the water. If the radius of the circle is increasing at the rate of meters per second, how fast is the area of the oil film growing when the radius is 100 meters? 4) ) If the radius of a sphere is increasing at the constant rate of 3mm sec, how fast is the volume changing when the surface area( 4π r ) is 10 square millimeters? 5) What is the radius of an expanding circle at a moment when the rate of change of its area is numerically twice as large as the rate of change of its radius? 6) A plane flying parallel to the ground at a height of 4 km passes over a radar station. A short time later, the radar equipment reveals that the plane is 5 km away and that the distance between the plane and the station is increasing at a rate of 300 km per hour. At that moment, how fast is the plane moving horizontally? 7) A spherical snowball is melting (symmetrically) at a rate of 4π cubic cm per hour. How fast is the diameter changing when it 0 centimeters?

7 Related Rates DAY 7 1) A cylindrical tank of radius 10 feet is being filled with wheat at the rate of 100π cubic feet per minute. How fast is the depth of the wheat increasing? (The volume of a cylinder is π rh, where r is its radius and h its height.) ) A boat passes a fixed buoy at 9 A.M., heading due west at 3 mph. Another boat passes the same buoy at 10 A.M., heading due north at 5 mph. How fast is the distance between the boats changing at 11:30 A.M.? 3) Water is pouring into an inverted cone at the rate of π cubic meters per minute. The height of the cone is 10 m and the radius of its base is 5 m. How fast is the water level rising when the water stands 7.5 meters in the cone? 4) A boat is being pulled into a dock by a rope that passes through a ring on the bow of the boat. The dock is 8 feet higher than the bow ring. How fast is the boat approaching the dock when the length of rope between the dock and the boat is 10 feet, if the rope is being pulled in at the rate of 3 feet per second? 5) A trough is 10 feet long and has a cross section in the shape of an equilateral triangle feet on each 3 side. If water is being pumped in at a rate of 0 ft min, how fast is the water level rising when the water is 1 ft deep? 3 6) Sand is being poured onto a conical pile at the constant rate of 50 ft min. Frictional forces in the sand are such that the height of the pile is always equal to the radius of the base. How fast is the height of the pile increasing when the sand is 5 feet deep?

8 C AP ^n orchard has an average yield of 5 bushels per tree when there are at most 40 trees per acre. When there are more than 40 trees per acre, the averageyield per tree decreases by I U,rrt el per tree for every tree over ' 40. Find the number of trees per acre that will give the greatest yield per acre. A" open box is made from a rectangular piece of cardboard that is 8 feet by 3 leet, by cutting out four equal squares from the corners and then fotoing,.,p itt. flaps. What length of the side of a square will yield the box with the largest volume? 6^ manufacturer sells each of his TV sets for $85. The cost C (in dollars) of manufacturing and selling x TV sets per week is C x If at most 10,000 sets can be produced per week, how many sets should be made and sold to maximrze the weekly profit? f,il rni, example illustrates a problem faced by a firm that delivers its products by truck' The (} ;.* rr". ij"ra that as the truck;i rpr.a increases the operating cost (gasoline, oil, and so on) increases, whereas the driver cost goes down. what is ihe most economical speed at which the truck should be driven? The operating cost for a certain truck is estimated to be (tr-%) cents per mile when it is driven at a speed of v miles per hour. The driver is paid $14 per hour. what speed will minimize the cost of making a delivery to a city k miles away? Assume that the law restricts the speed to 40 ( v < 60 '

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