4.6 Optimization Problems Rita Korsunsky

Applications of Maxima and Minima The first derivative may be used to find the largest or smallest value of a function.

Let the first number second number Compare the product F(x) at Example 1: Find two positive numbers whose sum is 20 and whose product is as large as possible. Let the first number second number Compare the product F(x) at the endpoints and at x = 10: Maximum Answer: First number = 10, Second number = 20-10 = 10 100 10 20

Example 2: A rectangle is inscribed in a semicircle of radius 2 Example 2: A rectangle is inscribed in a semicircle of radius 2. What is the largest area the rectangle can have and its dimensions? Let Length ; Height

Example 3: A square sheet of tin 16 inches on a side is to make an open-top box by cutting a small square of tin from each corner and bending up the sides. How large a square should be cut from each corner to make the box have as large a volume as possible. x 16

Example 4 A circular cylindrical metal container, open at the top, is to have a capacity of 24 in.3 The cost of the material used for the bottom of the container is 15 cents per in.2, and that of the material used for the curved part is 5 cents per in.2 If there is no waste of material, find the dimensions that will minimize the cost of the material. Cost of container = 15 (area of base) + 5 (lateral area) Express C in one variable: Plug into C: --- h r 2 C’ min + Radius by height: 2 by 6

Example 5 Find the maximum volume of a right circular cylinder that can be inscribed in a cone of altitude 12 centimeters, and base radius 4 centimeters, if the axes of the cylinder and cone coincide. Express V in terms of one variable: Plug into V: --- 12 r 4-r . + - h

Example 6 A north-south bridle path intersects an east-west river at point O. At noon, a horse and rider leave O traveling north at 12km/h. At the same time, a boat is 25km east of O traveling west at 16km/h. Express the distance d between the horse and the boat as a function of the time t in hours after noon. Find the time t when the horse and the boat are closest to each other and the minimum distance. Find the minimum of the function using d’(t) : . - + 1 Horse and boat are closest 1h after noon Minimum distance is 15km