Explanation Given: The field is enclosed by 600 m of fencing and no fencing is needed along one side of the field. Definition used: Lagrange Multipliers 1. The form of the constraint is g ( x , y ) = 0 . 2. The form of the Lagrange function F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . 3 Form the system of equations F x ( x , y , λ ) = 0 , F y ( x , y , λ ) = 0 and F λ ( x , y , λ ) = 0 , compute the solution of these system also the relative extrema among these solution. Calculation: Let x and y be the dimension of the field. Since the field is enclosed by 600 m of fencing and no fencing is needed along one side of the field, x + 2 y = 600 . The function f ( x , y ) = x y subject to the constraint x + 2 y = 600 . The constrain in the form g ( x , y ) = 0 . Thus, the given constrain can be written as g ( x , y ) = x + 2 y − 600 . The form of the Lagrange function is, F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . Substitute f ( x , y ) = x y and g ( x , y ) = x + 2 y − 600 in F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) , F ( x , y , λ ) = x y − λ ( x + 2 y − 600 ) = x y − x λ − 2 y λ + 600 λ Thus, the Lagrange function is F ( x , y , λ ) = x y − x λ − 2 y λ + 600 λ . Here, need to find the partial derivative of the function F ( x , y , λ ) . The partial derivative of the function F ( x , y , λ ) with respect to x and fix y and λ are constant. F x ( x , y , λ ) = ∂ ∂ x ( x y − x λ − 2 y λ + 600 λ ) = ∂ ∂ x ( x y ) − ∂ ∂ x ( x λ ) − ∂ ∂ x ( 2 y λ ) + ∂ ∂ x ( 600 λ ) = ( y ) ∂ ∂ x ( x ) − ( λ ) ∂ ∂ x ( x ) − ∂ ∂ x ( 2 y λ ) + ∂ ∂ x ( 600 λ ) Simplify the terms and obtain the partial derivatives. F x ( x , y , λ ) = ( y ) ( 1 ) − ( λ ) ( ) − 0 + 0 = y − λ Therefore, the partial derivative with respect to x is F x ( x , y , λ ) = y − λ . The partial derivative of the function F ( x , y , λ ) with respect to y and fix x and λ are constant. F y ( x , y , λ ) = ∂ ∂ y ( x y − x λ − 2 y λ + 600 λ ) = ∂ ∂ y ( x y ) − ∂ ∂ y ( x λ ) − ∂ ∂ y ( 2 y λ ) + ∂ ∂ y ( 600 λ ) = ( x ) ∂ ∂ y ( y ) − ∂ ∂ y ( x λ ) − ( 2 λ ) ∂ ∂ y ( y ) + ∂ ∂ y ( 600 λ ) Simplify the terms and obtain the partial derivatives

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Chapter 17.4, Problem 34E

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Chapter 17.4, Problem 33E

Chapter 17.4, Problem 35E

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There are three options for investing $1150. The first earns 10% compounded annually, the second earns 10% compounded quarterly, and the third earns 10% compounded continuously. Find equations that model each investment growth and use a graphing utility to graph each model in the same viewing window over a 20-year period. Use the graph to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three investment? STEP 1: The formula for compound interest is A = nt = P(1 + − − ) n², where n is the number of compoundings per year, t is the number of years, r is the interest rate, P is the principal, and A is the amount (balance) after t years. For continuous compounding, the formula reduces to A = Pert Find r and n for each model, and use these values to write A in terms of t for each case. Annual Model r=0.10 A = Y(t) = 1150 (1.10)* n = 1 Quarterly Model r = 0.10 n = 4 A = Q(t) = 1150(1.025) 4t Continuous Model r=0.10 A = C(t) =…

Chapter 17 Solutions

EBK FINITE MATHEMATICS AND CALCULUS WIT

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The area of the largest rectangular field.

Solution Summary: The author calculates the area of the largest rectangular field using Lagrange Multipliers. Since the field is enclosed by 600 m of fencing, x+2y=600.

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Chapter 17 Solutions