Explanation Given Information: The fencing for left border costs 8 dollar per foot and fencing the lower border costs 2 dollar per foot. The border along the river does not need fencing. The budget for fencing is 400 dollars. 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: Consider the formula: Step 1: Identify the unknowns There are two unknowns left border x and lower border y of the right triangle. Step 2: Identify the objective functions: The area of the patch is given by: A = x y 2 Step 3: Identify the constraint: 8 x + 2 y = 400 Step 4: State the optimization problem: The given objective function is given by: Maximize the objective function A = x y 2 Step 5: State and solve the given optimization problem: The objective function is given by: A = x y 2 The constraint is given by: 8 x + 2 y = 400 Writing y in terms of x to get: Subtracting 8 x from both sides of the equation to get: 8 x + 2 y − 8 x = 400 − 8 x 2 y = 400 − 8 x Dividing both sides of the equation by 2 to get: 2 y 2 = 400 − 8 x 2 y = 200 − 4 x Substituting the value of y as 200 − 4 x in the objective function to get: A = x y 2 A = x ( 200 − 4 x ) 2 A = 200 x − 4 x 2 2 A = 100 x − 2 x 2 Since, the area cannot be negative or zero

Student Solutions Manual for Waner/Costenoble's Finite Math and Applied Calculus, 7th

7th Edition

ISBN: 9781337275972

Author: Waner, Stefan; Costenoble, Steven

Publisher: Cengage Learning

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Chapter 12.2, Problem 22E

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To calculate: The dimensions of the largest area of a right triangular garden. The fencing for left border costs 8 dollar per foot and fencing the lower border costs 2 dollar per foot. The border along the river does not need fencing. The budget for fencing is 400 dollars.

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Chapter 12.2, Problem 21E

Chapter 12.2, Problem 23E

Chapter 12 Solutions

Student Solutions Manual for Waner/Costenoble's Finite Math and Applied Calculus, 7th

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The dimensions of the largest area of a right triangular garden. The fencing for left border costs 8 dollar per foot and fencing the lower border costs 2 dollar per foot. The border along the river does not need fencing. The budget for fencing is 400 dollars.

Solution Summary: The author calculates the dimensions of the largest area of a right triangular garden.

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To calculate: The dimensions of the largest area of a right triangular garden. The fencing for left border costs 8 dollar per foot and fencing the lower border costs 2 dollar per foot. The border along the river does not need fencing. The budget for fencing is 400 dollars.

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