Answer:
The approximate length of the garden is at least 22 feet and at most 51 feet.
Step-by-step explanation:
A rectangular garden must have a perimeter of 145 feet.
Let x feet be the lengths of the garden. Then the width of the garden can be found using the perimeter:
[tex]P=2(\text{Length}+\text{Width})\\ \\145=2(x+\text{Width})\\ \\\text{Width}=72.5-x[/tex]
The area of the garden is
[tex]A=\text{Length}\cdot \text{Width}\\ \\A=x\cdot (72.5-x)[/tex]
The area must be at least 1,100 square feet, then
[tex]x(72.5-x)\ge 1,110\\ \\72.5x-x^2-1,100\ge 0\\ \\x^2-72.5x+1,100\le 0[/tex]
Solve this quadratic inequality:
[tex]D=(-72.5)^2-4\cdot 1,100=856.25\\ \\\sqrt{D}=25\sqrt{1.37}\\ \\x_1=\dfrac{72.5-25\sqrt{1.37}}{2}\approx 22\\ \\x_2=\dfrac{72.5+25\sqrt{1.37}}{2}\approx 51\\ \\(x-22)(x-51)\le 0\\ \\x\in [22,51][/tex]
