HW 4 6 Optimization Math 400! Name 1. Find two positive numbers whose sum is 9 if the product of one number and the square of the other number is the maimum possible product of the 2 numbers. 2. A farmer wants to fence a rectangular piece of land. One side of the rectangle plot will not have any fence on it as it fronts a river with a straight bank. The other 3 sides will use 500 feet of fencing to enclose the largest area possible. Find the length and width of the rectangular plot and the maimum area it encloses.! HW 4 6! Page 1 of 8! 201 8 Eitel
3. A farmer wants to make a rectangular pen that encloses a total area of 216 square feet. He wants 2 pens of equal size inside the rectangular pen as shown. Find the dimensions of the outside rectangular pen and the minimum amount of fencing needed. 4. A farmer wants to make a rectangular pen using a total of 500 feet of fencing. He wants to have three parallel partitions inside the pen as shown below. Find the dimensions of the outside rectangular pen that will maimize the total area of the pen? HW 4 6! Page 2 of 8! 201 8 Eitel
5. A piece of cardboard is 30 inches by 30 inches. A square is cut out of each corner as shown below. The sides of the remaining shape are then folded up to form a bo. Determine the height of the bo that will give a maimum volume. 30 30 6. A piece of cardboard is 8 inches by 15 inches. A square is cut out of each corner as shown below. The sides of the remaining shape are then folded up to form a bo. Determine the height of the bo that will give a maimum volume. 15 8 HW 4 6! Page 3 of 8! 201 8 Eitel
7. You want to make the bo shown below with an open top and a square base that contains the maimum volume using 48 square cm of material for the sides and bottom. What are the dimensions of the bo and what is the largest volume the bo can contain? 8. You want to make a tank with a rectangular bottom and rectangular sides that will be open at the top. It is constructed so its width is 4 meters and its volume is 36 cubic meters. The materials to build the tank cost $10 per square meter for the bottom and $ 5 per square meter for the sides. Find the cost of the least epensive tank? HW 4 6! Page 4 of 8! 201 8 Eitel
9. A landscape architect plans to enclose a 3000 square foot rectangular region in a botanical garden, She will use shrubs costing $ 15 per foot along three sides and fencing costing $10 per foot along the fourth side. Find the minimum total cost. y fence shrubs shrubs y shrubs 10. A container in the shape of a right circular cylinder with no top has an outside surface area of 3π square feet. What height h and base radius r will maimize the volume of the cylinder? HW 4 6! Page 5 of 8! 201 8 Eitel
11. A cylindrical can is to hold 20π cubic meters. The material for the top and bottom costs $10 a square meter and material for the side costs $ 8 a square meter. Find the radius r and height h of the can with the minimum cost. 12. Find the minimum distance from the point ( 3, 0( and the parabola y = 2. D = ( 1 ) 2 + ( y y 1 ) 2 HW 4 6! Page 6 of 8! 201 8 Eitel
13. A Florida citrus grower has 60 orange trees planted and his average yield per tree is 400 oranges for a yield of 24000 oranges. For each new tree he planes the average yield will decrease by 4 oranges per tree. How many new trees should he plants to maimize the total yield. Yield = number of trees * yield per tree. let n be the number of new trees added. 14. The city of New York charges $10 for a one day ticket to park in a parking space. It is averaging 27,000 ticket sales a day. A recent research paper concluded that for each 10 cents decrease in price the number of new tickets sold would increase by 300. What should the ticket price be sat at to maimize revenue? Revenue = price * number of tickets sold. Let n = the number of times the price is reduces by 10 cents. HW 4 6! Page 7 of 8! 201 8 Eitel
15. A pipe line needs to be run from a refinery on one side of a river to a storage tank on the other side of the river, The refinery is 8 km downriver from the tank and the river is 4 km wide. It costs $ 5,000 a km to run the pipeline under the river and $ 3,000 a km to run along the river bank. Find the distance () where the pipeline should cross the river to minimize the cost of construction. Use a calculator to find a decimal answer rounded off a whole number refinery 8 km y 2 km storage tank HW 4 6! Page 8 of 8! 201 8 Eitel