The length and width of maximum possible garden are 12.404 feet and 4.702 feet, respectively.
Procedure - Determination of the largest possible garden for a given budget
By geometry we know that the area of a rectangle ([tex]A[/tex]), in square feet, is equal to the product of the length and the width of the garden, that is:
[tex]A = (2\cdot x + 3) \cdot x[/tex] (1)
And the cost function ([tex]C[/tex]), in monetary units, is the product of the fencing costs ([tex]c[/tex]), in monetary units per feet, and the area of the garden ([tex]A[/tex]):
[tex]C = c\cdot A[/tex] (2)
Now we proceed to perform first and second derivative tests to the area of the rectangle:
First derivative tests
[tex]4\cdot x + 3 = 0[/tex] (3)
Second derivative tests
[tex]A'' = 4[/tex] (4)
By (4) we know that only a relative minimum exists and we must determine a possible maximum by analyzing (1) and (2):
[tex]A = \frac{C}{c}[/tex]
[tex]\frac{C}{c} = (2\cdot x + 3)\cdot x[/tex]
[tex]\frac{C}{c} = 2\cdot x ^{2} + 3\cdot x[/tex]
If we know that [tex]C = 175[/tex] and [tex]c = 3[/tex], then the length and the width of the maximum possible garden are:
[tex]2\cdot x^{2}+3\cdot x -58.333 = 0[/tex] (5)
And the solution of this second order polynomial are determined by quadratic formula:
[tex]x_{1} \approx 4.702[/tex], [tex]x_{2} \approx -6.202[/tex]
The only 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. [tex]\blacksquare[/tex]
Remark
The statement is incomplete, complete form is presented below:
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 [tex]2\cdot x + 3[/tex] feet and a width of [tex]x[/tex] feet. Fencing material costs $ 3 per foot.
What are the largest possible dimensions of a garden that Mr. Jones could build with a budget of $ 175?
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