A rectangular garden must have a perimeter of 145 feet and an area of at least 1100 square feet. Describe the possible lengths of the garden. Round your answers to the nearest foot. The approximate length of the garden is at least feet and at most feet.

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]