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):
![A = \frac{C}{c}](https://tex.z-dn.net/?f=A%20%3D%20%5Cfrac%7BC%7D%7Bc%7D)
![\frac{C}{c} = (2\cdot x + 3)\cdot x](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7Bc%7D%20%3D%20%282%5Ccdot%20x%20%2B%203%29%5Ccdot%20x)
![\frac{C}{c} = 2\cdot x ^{2} + 3\cdot x](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7Bc%7D%20%20%3D%202%5Ccdot%20x%20%5E%7B2%7D%20%2B%203%5Ccdot%20x)
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:
, ![x_{2} \approx -6.202](https://tex.z-dn.net/?f=x_%7B2%7D%20%5Capprox%20-6.202)
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. ![\blacksquare](https://tex.z-dn.net/?f=%5Cblacksquare)
<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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