The length and width of maximum possible garden are 12.404 feet and 4.702 feet, respectively.
<h3>Procedure - Determination of the largest possible garden for a given budget</h3>
By geometry we know that the area of a rectangle (), in square feet, is equal to the product of the length and the width of the garden, that is:
(1)
And the cost function (), in monetary units, is the product of the fencing costs (), in monetary units per feet, and the area of the garden ():
(2)
Now we proceed to perform first and second derivative tests to the area of the rectangle:
<h3>First derivative tests</h3>
(3)
<h3>Second derivative tests</h3>
(4)
By (4) we know that only a <em>relative</em> minimum exists and we must determine a possible maximum by analyzing (1) and (2):
If we know that and , then the length and the width of the maximum possible garden are:
(5)
And the solution of this <em>second order</em> polynomial are determined by quadratic formula:
,
The <em>only</em> root that is mathematically and physically reasonable is approximately 4.702 feet, and the length and width of maximum possible garden are 12.404 feet and 4.702 feet, respectively.
<h3>Remark</h3>
The statement is incomplete, complete form is presented below:
<em>Mr. Jones is going to build a garden in back of the restaurant to have fresh produce available. The garden will be rectangular, with a length of </em><em> feet and a width of </em><em> feet. Fencing material costs $ 3 per foot. </em>
<em />
<em>What are the largest possible dimensions of a garden that Mr. Jones could build with a budget of $ 175?</em>
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