Eg.1: Kara and James are both training for a marathon. Kara's house is located 35 km north of James house. At 6:00 a.m. on a Monday morning, Kara leaves her house and jogs South at 1 1 km/h. At the same time, James leaves his house and jogs East at 8 km/h. When are Kara and James closest together, given that they both run for 2.5 hours?Eg.3: Cindy makes a candle holder by inscribing a cylinder in a cone. The height of the cone is 15 cm. The radius is 5 cm. Find the dimensions of the cylinder that will maximize its surface area? Radlus 0f cone AACD and AABE are similar triangles because angle A is common to both triangles and angle C and angle B are both 90" therefore angle D h 15 B E and angle E must be equal due to triangle sum. 15-h Since the two triangles are similar their sides are proportional. Radius of cylinder 1e: i5: = Q = Big vs. Small A In the cone AB BE AE This allows us to now express the surface area as a function of the radius. Therefore the Surface Area of the cylinder is maximized when the radius is 15M cm and the height is 15M cm. (3.75 cm) Eg. l: A rectangular piece of land is to be fenced using two kinds of fencing. Two opposite sides will be fenced using standard fencing with a cost of $6/m, while the other 2 require heavy duty fencing that cost $9/m. What are the dimensions of the rectangular lot of greatest area that can be fenced for a cost of $9000. A railroad between two cities carries 10 000 passengers per year when the fare is $50. If the fare goes up, the number of passengers will decrease, since more people will drive. It is estimated that each $10 increase in fare will result in 1000 fewer passengers per year. The train can not carry more than 13000 passengers per year. a) At what fare will the train have no passengers? b) What fare will maximize revenue? c) When revenue is maximized, how many people will ride the train each year? Eg.l: A cylindrical chemical storage tank with a capacity of 1000 m3 is going to be constructed in a warehouse that is 12 m by 15 m by 11 m in height, The specifications call for the base to be made of sheet steel that costs $100/rn2, the top to be made of sheet steel that costs $50/m2 and the wall to be made of sheet steel that costs $80/m2. a) Determine whether it is possible for a tank of this capacity to fit in the warehouse. If it is possible, state the restrictions on the radius of the tank. b) If fitting the tank in the warehouse is possible, determine the proportions that meet the conditions and that minimize the cost of the steel for construction
