Do it on paper! DO IT ON PAPER PLEASE!!!

Optimization Problems Day # 1 Optimization is finding the maximum or minimum quantity of a function under certain circumstances or restraints. Steps for solving optimization problems: 1. Understand the problem. a. Identify the quantity that can vary. b. Express the quantity to be optimized by a function with only one variable. 2. Draw and label a diagram, whenever possible. 3. Determine the domain of the function (ie: restrictions on the variable that we are considering) 4. Find all the extreme values. Eg.l: A carpenter is building an open box with a square base for holding firewood. The box must have a surface area of 8 m2. Wha dimensions will yield a maximum volume? Eg.2: A piece of cardboard which measures 24 cm by 24 cm is to be made into a box by cutting squares out of the corners and folding up the flaps. What dimensions result in a box with a maximum volume? What is this volume? Unit 3 - Activity MCV4U Optimization Problems 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 11 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.2: Find the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of lengths 5 cm and 12 cm. The two sides of the rectangle lie along the legs. Optimization Problems 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? Rad'us 0f cene 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. 15h Since the two triangles are similar their sides are proportional. Radius of cylinder 1e: = Q = 31 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. This allows us to new 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 154 cm. (3.75 cm]