Math 180 Worksheets Wii1 11 Optimization, Linear Approximation Keywords: optimization, word problems, linear approzimation For the problems below, include the domain of the function to be optimized, as well as an argument why your critical point(s) yields an absolute maximum or minimum. 1. In your own words, describe what is the goal of an optimization problem, without using the word optimize. 2. A three sided fence is to be built next to a straight river which forms the fourth side of a rectangular region. The enclosed area is to equal 3200 ft2. The cost of the fence is $2/ft. This question will have you find the dimensions that minimize the cost of the fence. (a) What quantity needs to be optimized? (b) Draw a picture modeling this situation. () Find a function for the quantity that is being optimized and find its domain. (d) Find the dimensions that minimize the cost of the fence. 34 Math 180 Worksheets Wi1 3. What two non-negative real numbers @ and b whose sum is 23 maximize a? + b*? Minimize a? 4+ #?? Solve both questions using optimization techniques even if you know the answer. 4. A square-based, box-shaped shipping crate is designed to have a volume of 16 ft*. The material used to make the base costs twice as much (per square foot) as the material used in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimensions of the crate that minimize the cost of the materials? Math 180 Worksheets Wii1 5. Find the linear approximation L(x) of the function f(r) = In(r) at a = 1 and use it to approximate In0.9. 6. A piece of wire of length 60 cm is cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to minimize the combined area of the circle and the square? 36