What route will take the least amount of time ?Ĭlick HERE to see a detailed solution to problem 9. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim. You must first swim across the river to any point on the opposite bank. PROBLEM 9 : You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river.2 Find the radius r and height h of the most economical can.Ĭlick HERE to see a detailed solution to problem 8. 3 The material for the top and bottom costs $10/m. PROBLEM 8 : A cylindrical can is to hold 20.PROBLEM 7 : Find the point (x, y) on the graph ofĬlick HERE to see a detailed solution to problem 7.Find the dimensions of the triangle with the shortest hypotenuse.Ĭlick HERE to see a detailed solution to problem 6. PROBLEM 6 : Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes.What will be the dimensions of the box with largest volume ?Ĭlick HERE to see a detailed solution to problem 5. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. PROBLEM 5 : A sheet of cardboard 3 ft.2 What height h and base radius r will maximize the volume of the cylinder ?Ĭlick HERE to see a detailed solution to problem 4. PROBLEM 4 : A container in the shape of a right circular cylinder with no top has surface area 3įt.What dimensions will result in a box with the largest possible volume ?Ĭlick HERE to see a detailed solution to problem 3. PROBLEM 3 : An open rectangular box with square base is to be made from 48 ft.What dimensions will maximize the total area of the pen ?Ĭlick HERE to see a detailed solution to problem 2. PROBLEM 2 : Build a rectangular pen with three parallel partitions using 500 feet of fencing.PROBLEM 1 : Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.Ĭlick HERE to see a detailed solution to problem 1.The following problems range in difficulty from average to challenging. Verify that your result is a maximum or minimum value using the first or second derivative test for extrema. Then differentiate using the well-known rules of differentiation.Ħ. Before differentiating, make sure that the optimization equation is a function of only one variable. Some problems may have two or more constraint equations.ĥ. Some problems may have NO constraint equation. This is then substituted into the "optimization" equation before differentiation occurs. The "constraint" equation is used to solve for one of the variables. One equation is a "constraint" equation and the other is the "optimization" equation. Experience will show you that MOST optimization problems will begin with two equations. Clearly denote that equation which you are asked to maximize or minimize. Write down all equations which are related to your problem or diagram. This step is very important because it leads directly or indirectly to the creation of mathematical equations.Ĥ. Define variables to be used and carefully label your picture or diagram with these variables. Pictures are a great help in organizing and sorting out your thoughts.ģ. If appropriate, draw a sketch or diagram of the problem to be solved. If you misread the problem or hurry through it, you have NO chance of solving it correctly.Ģ. It is imperative to know exactly what the problem is asking. Read the problem at least three times before trying to solve it. However, if you are patient you can minimize your anxiety and maximize your success with these problems by following these guidelines : Many students find these problems intimidating because they are "word" problems, and because there does not appear to be a pattern to these problems. They illustrate one of the most important applications of the first derivative. The following problems are maximum/minimum optimization problems.
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