Math  /  Geometry

QuestionQuestion A rectangular garden is to be constructed using a river as one side of the garden and wood fencing for the other three sides. Given 144 ft of wood fencing, what is the maximum area the garden can be?
Provide your answer below:

Studdy Solution
Verify the solution by checking the second derivative or using a test:
The second derivative is:
A(x)=ddx(1442x2) A''(x) = \frac{d}{dx}\left(\frac{144 - 2x}{2}\right) =1 = -1
Since A(x)<0 A''(x) < 0 , the function A(x) A(x) has a maximum at x=72 x = 72 .
Calculate y y when x=72 x = 72 :
y=144722=36 y = \frac{144 - 72}{2} = 36
Thus, the maximum area is:
A(72)=72×36=2592 square feet A(72) = 72 \times 36 = 2592 \text{ square feet}
The maximum area of the garden is:
2592 square feet \boxed{2592 \text{ square feet}}

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