a. Answer A = 8 x ( 50 − x ) 3 Explanation Calculation: Two adjacent rectangular corrals given below are enclosed by 200 feet of fencing. The perimeter of the corrals is: P = 4 x + 3 y It is given that the corrals are enclosed by 200 feet of fencing which is equal to its perimeter. This gives that, P = 200 4 x + 3 y = 200 3 y = 200 − 4 x y = 200 − 4 x 3 Now area of the corrals is, Area = Length × Width A = 2 x × y = 2 x y Substitute y = 200 − 4 x 3 in the above area. A = 2 x y = 2 x ( 200 − 4 x 3 ) = 8 x ( 50 − x ) 3 Conclusion The area A of the corrals as a function of x is A = 8 x ( 50 − x ) 3 . b. To determine To find: the dimensions that produce maximum enclosed area. Answer The dimensions that produce maximum enclosed area are x = 25 ft and y = 33 1 3 ft . Explanation Given: The area A of the corrals as a function of x is A = 8 x ( 50 − x ) 3 = 400 3 x − 8 3 x 2 Concept Used: For the function f ( x ) = a x 2 + b x + c with vertex ( − b 2 a , f ( − b 2 a ) ) , When a > 0 , f has a minimum at x = − b 2 a and the minimum value is f ( − b 2 a ) . When a < 0 , f has a maximum at x = − b 2 a and the maximum value is f ( − b 2 a ) . Calculation: Now compare the given function A = 400 3 x − 8 3 x 2 with f ( x ) = a x 2 + b x + c . Clearly a = − 8 3 < 0 and b = 400 3 So by the above definition, f has a maximum at x = − b 2 a and the maximum value is f ( − b 2 a ) . So, the dimensions that produce maximum enclosed area is, x = − b 2 a = − 400 3 2 × ( − 8 3 ) = 25 From part (a), y = 200 − 4 ( 25 ) 3 = 200 − 100 3 = 100 3 = 33 1 3 Conclusion: The dimensions that produce maximum enclosed area are x = 25 ft and y = 33 1 3 ft .

4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Question

Chapter 2.1, Problem 73E

To determine

To find: the area A of the corrals as a function of x .

Expert Solution

Answer to Problem 73E

A=8x(50−x)3

Explanation of Solution

Calculation:

Two adjacent rectangular corrals given below are enclosed by 200 feet of fencing.

EBK PRECALCULUS W/LIMITS, Chapter 2.1, Problem 73E

The perimeter of the corrals is:

P=4x+3y

It is given that the corrals are enclosed by 200 feet of fencing which is equal to its perimeter. This gives that,

P=2004x+3y=2003y=200−4xy=200−4x3

Now area of the corrals is,

Area = Length × Width

A=2x×y=2xy

Substitute y=200−4x3 in the above area.

A=2xy=2x(200−4x3)=8x(50−x)3

Conclusion The area A of the corrals as a function of x is A=8x(50−x)3 .

To determine

To find: the dimensions that produce maximum enclosed area.

Expert Solution

Answer to Problem 73E

The dimensions that produce maximum enclosed area are x=25 ft and y=3313 ft .

Explanation of Solution

Given:

The area A of the corrals as a function of x is

A=8x(50−x)3=4003x−83x2

Concept Used:

For the function f(x)=ax2+bx+c with vertex (−b2a,f(−b2a)) ,

When a>0 , f has a minimum at x=−b2a and the minimum value is f(−b2a) .

When a<0 , f has a maximum at x=−b2a and the maximum value is f(−b2a) .

Calculation:

Now compare the given function A=4003x−83x2 with f(x)=ax2+bx+c .

Clearly a=−83<0 and b=4003

So by the above definition, f has a maximum at x=−b2a and the maximum value is f(−b2a) .

So, the dimensions that produce maximum enclosed area is,

x=−b2a=−40032×(−83)=25

From part (a),

y=200−4(25)3=200−1003=1003=3313

Conclusion:

The dimensions that produce maximum enclosed area are x=25 ft and y=3313 ft .

Chapter 2.1, Problem 72E

Chapter 2.1, Problem 74E

Chapter 2 Solutions

EBK PRECALCULUS W/LIMITS

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Prob. 64E Ch. 2.6 - Prob. 65E Ch. 2.6 - Prob. 66E Ch. 2.6 - Prob. 67E Ch. 2.6 - Prob. 68E Ch. 2.6 - Prob. 69E Ch. 2.6 - Prob. 70E Ch. 2.6 - Prob. 71E Ch. 2.6 - Prob. 72E Ch. 2.6 - Prob. 73E Ch. 2.6 - Prob. 74E Ch. 2.6 - Prob. 75E Ch. 2.6 - Prob. 76E Ch. 2.6 - Prob. 77E Ch. 2.6 - Prob. 78E Ch. 2.6 - Prob. 79E Ch. 2.6 - Prob. 80E Ch. 2.6 - Prob. 81E Ch. 2.6 - Prob. 82E Ch. 2.7 - Prob. 1E Ch. 2.7 - Prob. 2E Ch. 2.7 - Prob. 3E Ch. 2.7 - Prob. 4E Ch. 2.7 - Prob. 5E Ch. 2.7 - Prob. 6E Ch. 2.7 - Prob. 7E Ch. 2.7 - Prob. 8E Ch. 2.7 - Prob. 9E Ch. 2.7 - Prob. 10E Ch. 2.7 - Prob. 11E Ch. 2.7 - Prob. 12E Ch. 2.7 - Prob. 13E Ch. 2.7 - Prob. 14E Ch. 2.7 - Prob. 15E Ch. 2.7 - Prob. 16E Ch. 2.7 - Prob. 17E Ch. 2.7 - Prob. 18E Ch. 2.7 - Prob. 19E Ch. 2.7 - Prob. 20E Ch. 2.7 - Prob. 21E Ch. 2.7 - Prob. 22E Ch. 2.7 - Prob. 23E Ch. 2.7 - Prob. 24E Ch. 2.7 - Prob. 25E Ch. 2.7 - Prob. 26E Ch. 2.7 - Prob. 27E Ch. 2.7 - Prob. 28E Ch. 2.7 - Prob. 29E Ch. 2.7 - Prob. 30E Ch. 2.7 - Prob. 31E Ch. 2.7 - Prob. 32E Ch. 2.7 - Prob. 33E Ch. 2.7 - Prob. 34E Ch. 2.7 - Prob. 35E Ch. 2.7 - Prob. 36E Ch. 2.7 - Prob. 37E Ch. 2.7 - Prob. 38E Ch. 2.7 - Prob. 39E Ch. 2.7 - Prob. 40E Ch. 2.7 - Prob. 41E Ch. 2.7 - Prob. 42E Ch. 2.7 - Prob. 43E Ch. 2.7 - Prob. 44E Ch. 2.7 - Prob. 45E Ch. 2.7 - Prob. 46E Ch. 2.7 - Prob. 47E Ch. 2.7 - Prob. 48E Ch. 2.7 - Prob. 49E Ch. 2.7 - Prob. 50E Ch. 2.7 - Prob. 51E Ch. 2.7 - Prob. 52E Ch. 2.7 - Prob. 53E Ch. 2.7 - Prob. 54E Ch. 2.7 - Prob. 55E Ch. 2.7 - Prob. 56E Ch. 2.7 - Prob. 57E Ch. 2.7 - Prob. 58E Ch. 2.7 - Prob. 59E Ch. 2.7 - Prob. 60E Ch. 2.7 - Prob. 61E Ch. 2.7 - Prob. 62E Ch. 2.7 - Prob. 63E Ch. 2.7 - Prob. 64E Ch. 2.7 - Prob. 65E Ch. 2.7 - Prob. 66E Ch. 2.7 - Prob. 67E Ch. 2.7 - Prob. 68E Ch. 2.7 - Prob. 69E Ch. 2.7 - Prob. 70E Ch. 2.7 - Prob. 71E Ch. 2.7 - Prob. 72E Ch. 2.7 - Prob. 73E Ch. 2.7 - Prob. 74E Ch. 2.7 - Prob. 75E Ch. 2.7 - Prob. 76E Ch. 2.7 - Prob. 77E Ch. 2.7 - Prob. 78E Ch. 2.7 - Prob. 79E Ch. 2.7 - Prob. 80E Ch. 2.7 - Prob. 81E Ch. 2.7 - Prob. 82E Ch. 2.7 - Prob. 83E Ch. 2.7 - Prob. 84E Ch. 2.7 - Prob. 85E Ch. 2.7 - Prob. 86E Ch. 2.7 - Prob. 87E Ch. 2.7 - Prob. 88E Ch. 2.7 - Prob. 89E Ch. 2.7 - Prob. 90E Ch. 2 - Prob. 1RE Ch. 2 - Prob. 2RE Ch. 2 - Prob. 3RE Ch. 2 - Prob. 4RE Ch. 2 - Prob. 5RE Ch. 2 - Prob. 6RE Ch. 2 - Prob. 7RE Ch. 2 - Prob. 8RE Ch. 2 - Prob. 9RE Ch. 2 - Prob. 10RE Ch. 2 - Prob. 11RE Ch. 2 - Prob. 12RE Ch. 2 - Prob. 13RE Ch. 2 - Prob. 14RE Ch. 2 - Prob. 15RE Ch. 2 - Prob. 16RE Ch. 2 - Prob. 17RE Ch. 2 - Prob. 18RE Ch. 2 - Prob. 19RE Ch. 2 - Prob. 20RE Ch. 2 - Prob. 21RE Ch. 2 - Prob. 22RE Ch. 2 - Prob. 23RE Ch. 2 - Prob. 24RE Ch. 2 - Prob. 25RE Ch. 2 - Prob. 26RE Ch. 2 - Prob. 27RE Ch. 2 - Prob. 28RE Ch. 2 - Prob. 29RE Ch. 2 - Prob. 30RE Ch. 2 - Prob. 31RE Ch. 2 - Prob. 32RE Ch. 2 - Prob. 33RE Ch. 2 - Prob. 34RE Ch. 2 - Prob. 35RE Ch. 2 - Prob. 36RE Ch. 2 - Prob. 37RE Ch. 2 - Prob. 38RE Ch. 2 - Prob. 39RE Ch. 2 - Prob. 40RE Ch. 2 - Prob. 41RE Ch. 2 - Prob. 42RE Ch. 2 - Prob. 43RE Ch. 2 - Prob. 44RE Ch. 2 - Prob. 45RE Ch. 2 - Prob. 46RE Ch. 2 - Prob. 47RE Ch. 2 - Prob. 48RE Ch. 2 - Prob. 49RE Ch. 2 - Prob. 50RE Ch. 2 - Prob. 51RE Ch. 2 - Prob. 52RE Ch. 2 - Prob. 53RE Ch. 2 - Prob. 54RE Ch. 2 - Prob. 55RE Ch. 2 - Prob. 56RE Ch. 2 - Prob. 57RE Ch. 2 - Prob. 58RE Ch. 2 - Prob. 59RE Ch. 2 - Prob. 60RE Ch. 2 - Prob. 61RE Ch. 2 - Prob. 62RE Ch. 2 - Prob. 63RE Ch. 2 - Prob. 64RE Ch. 2 - Prob. 65RE Ch. 2 - Prob. 66RE Ch. 2 - Prob. 67RE Ch. 2 - Prob. 68RE Ch. 2 - Prob. 1CT Ch. 2 - Prob. 2CT Ch. 2 - Prob. 3CT Ch. 2 - Prob. 4CT Ch. 2 - Prob. 5CT Ch. 2 - Prob. 6CT Ch. 2 - Prob. 7CT Ch. 2 - Prob. 8CT Ch. 2 - Prob. 9CT Ch. 2 - Prob. 10CT Ch. 2 - Prob. 11CT Ch. 2 - Prob. 12CT Ch. 2 - Prob. 13CT Ch. 2 - Prob. 14CT Ch. 2 - Prob. 15CT Ch. 2 - Prob. 16CT Ch. 2 - Prob. 17CT Ch. 2 - Prob. 18CT Ch. 2 - Prob. 1PS Ch. 2 - Prob. 2PS Ch. 2 - Prob. 3PS Ch. 2 - Prob. 4PS Ch. 2 - Prob. 5PS Ch. 2 - Prob. 6PS Ch. 2 - Prob. 7PS Ch. 2 - Prob. 8PS Ch. 2 - Prob. 9PS Ch. 2 - Prob. 10PS Ch. 2 - Prob. 11PS Ch. 2 - Prob. 12PS Ch. 2 - Prob. 13PS

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