MCV 4U0 MORE MAX/MIN PROBLEMS 1. A page with printed area of 378 cm2 has top and side margins of 3 cm and a bottom margin of 4 cm. Find the dimensions of the entire page so that its area will be a minimum. 2. Waleed is in a rowboat 400 m offshore from a long, straight beach. He wishes to go to a point on the beach 1000 m further along. If he can walk at 3 km/h and row at 2 km/h, where should he beach his rowboat to reach his destination at the earliest time? 3. A lighthouse, L, is located on a small island 4 km west of point A on a straight north-south coastline. A power cable is to be laid from L to the nearest source of power at point B on the shoreline, 12 km north of point A. The cost of laying cable under water is $6000/km and the cost of laying cable along the shoreline is $2000/km. What length of cable should be laid along the shoreline? 4. A health drink manufacturer wants to sell its product in 400 mL cans. The metal used for the top and bottom of a can costs $1.50/m 2 and the metal used for the side costs $0.50/m2. The metal left over after the circles for the top and bottom of one can are cut out of one rectangle will be scrapped. Find the dimensions of the can that will minimize the cost of materials. 5. A closed rectangular box with a square base will be made as follows. The volume must be 0.9 m3. The area of the base must not exceed 1.8 m 2. The height of the box must not exceed 0.75 m. Determine the dimensions for a) minimal surface area and b) maximum surface area. 6. Moneeza is having a wooden box made (to hold her scarves and other fancy accessories!). It will be rectangular in shape with its length (along the front) twice as long as its width. The top, front, and two sides of the box are made of oak; the back and the bottom are made of pine. The box is to have a volume of 0.25m 3. If oak costs three times as much as pine, find the dimensions that minimize the cost of the box. 7. 3 A truck burns fuel at a rate of 0.002x + litres per kilometre when its constant x speed is x km/h. Fuel costs $0.73/L and the truck driver earns $15/h. What steady speed will minimize the cost of driving the truck if the truck must be driven 150 km? 8. A dune buggy is on a straight, north-south, desert road, 40 km north of Dustin City. The vehicle can travel at 75 km/h on the road and 45 km/h off the road. The driver wants to get to Gulch City, 50 km east of Dustin City by another straight road, in the shortest possible time. Determine the route he should take. 9. Corn silos are usually in the shape of a cylinder surmounted by a hemisphere. If the average yield on a given farm requires that the silo contain 1000 m 3 of corn, what radius of the silo would use the minimum amount of materials? Answers: 1. 24 cm X 28 cm 2. 358 m along the beach 3. 10 586 m 4. r = 2.55 cm, h = 19.58 cm 5. a) 1.2 1.2 0.75 m , b) 1.8 1.8 0.5 m 6. w = 0.478 m, l = 0.956 m, h = 0.547 m 7. 108.5 km/h 8. off road to a point 20 km west of Gulch City 9. r=3 600 m