Explanation Given Information: The provided expression is: P = 2 ( x + y ) Formula used: Steps to solve optimization problems: Step 1: Identify the unknowns with the aid of diagram. Step 2: Identify the objective function, the quantity to maximize or minimize. Step 3: Identify the constraint, the equations relating variables or inequalities expressing limitations on the value of the variables. Step 4: State the optimization problem Step 5: Eliminate the extra variables, solve the constraint for one unknown and substitute in the objective function. Step 6: Find the absolute maximum or absolute minimum of the objective function. Calculation: Let x be the length and y be the breadth. To find the dimensions at which the area will be smallest, we have to find the minimization of the area. The minimization of the perimeter of rectangle with length x and breadth y , P = 2 ( x + y ) subject to x y = 100 when x , y , > 0 . Consider the provided expression: P = 2 ( x + y ) Step 1: Identify the unknowns The unknowns are x and y . Step 2: Identify the objective functions: The objective function is given by: P = 2 ( x + y ) Step 3: Identify the constraint: The constraint is given by: x y = 100 Step 4: State the optimization problem: The given objective function is given by: Minimize the objective function P = 2 ( x + y ) The constraint is given by: x y = 100 , x , y > 0 Step 5: Eliminate the extra variables, solve the constraint for one unknown and substitute in the objective function. The constraints are given by x y = 100 Solving the given constraint for x with respect to y to get: x y = 100 x = 100 y Substituting 100 y for x in the objective function to get: P = 2 ( x + y ) P = 2 ( 100 y + y ) Since x > 0 Thus, 100 y > 0 100 > y Thus, 0 < y < 100 Minimizing the objective function subject to 0 < y < 100 to get: P = 2 ( 100 y + y ) Differentiating both sides of the equation with respect to y to get: P ' ( y ) = 2 ( − 100 y 2 + 1 ) Locating singular points to get: P ' ( y ) = 0 2 ( − 100 y 2 + 1 ) = 0 Multiplying by 1 2 to both sides of the equation to get: 2 ( − 100 y 2 + 1 ) 1 2 = 0 − 100 y 2 +

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 12.2, Problem 10E

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To calculate: The dimension of a rectangle at which its area will be smallest as 100.

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Chapter 12.2, Problem 9E

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The dimension of a rectangle at which its area will be smallest as 100.

Solution Summary: The author explains the steps to solve optimization problems: Identify the unknowns, identify the objective function, and find the absolute maximum or absolute minimum.

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