Chapter 4 | Applications of Derivatives
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4.31 Determine the maximum area if we want to make the same rectangular garden as in Figure 4.63 , but we have 200 ft of fencing.
Now let’s look at a general strategy for solving optimization problems similar to Example 4.32 .
Problem-Solving Strategy: Solving Optimization Problems 1. Introduce all variables. If applicable, draw a figure and label all variables. 2. Determine which quantity is to be maximized or minimized, and for what range of values of the other variables (if this can be determined at this time). 3. Write a formula for the quantity to be maximized or minimized in terms of the variables. This formula may involve more than one variable. 4. Write any equations relating the independent variables in the formula from step 3. Use these equations to write the quantity to be maximized or minimized as a function of one variable. 5. Identify the domain of consideration for the function in step 4 based on the physical problem to be solved. 6. Locate the maximum or minimum value of the function from step 4. This step typically involves looking for critical points and evaluating a function at endpoints. Now let’s apply this strategy to maximize the volume of an open-top box given a constraint on the amount of material to be used. Example 4.33 Maximizing the Volume of a Box An open-top box is to be made from a 24 in. by 36 in. piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. What size square should be cut out of each corner to get a box with the maximum volume? Solution Step 1: Let x be the side length of the square to be removed from each corner ( Figure 4.64 ). Then, the remaining four flaps can be folded up to form an open-top box. Let V be the volume of the resulting box.
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