Eg.1: 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 comers and folding up the aps. What dimensions result in a box with a maximum volume? What is this volume? 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. Eg.3: Cindy makes a candle holder by inscrihing 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? Radius 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 Ie' 9 = Q = A_D 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 15/4 cm and the height is 15M cm. (3.75 cm)