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Lea Anne is fencing a rectangular plot for her garden. She has 100 feet of fencing available. If she wants the plot to have an area of at least 400 square feet, what are the possible lengths
of the plot?
A The length should be longer than 4.2 feet but shorter than 95.8 feet.
B. The length should be either shorter than 4.2 feet or longer than 95.8 feet.
C. The length should be longer than 10 feet but shorter than 40 feet.
D. The length should be either shorter than 10 feet or longer than 40 feet

Respuesta :

Answer:

C. The length should be longer than 10 feet but shorter than 40 feet.

Step-by-step explanation:

To find the possible lengths of the plot, we create the system of equations based on the given information.

Let:

  • l = length of the plot
  • w = width of the plot

Given:

  • "She has 100 feet of fencing available", which means the perimeter of the rectangular plot = 100 feet → [tex]\boxed{2(l+w)=100}[/tex]
  • "an area of at least 400 square feet", which means the area of the rectangular plot ≥ 400 ft² → [tex]\boxed{l\times w\geq 400}[/tex]

Now, we combine both equations to find the length of the plot:

[tex]\displaystyle\left \{{{2(l+w)=100}\atop{l\times w\geq 400}} \right.[/tex]

Since we want to keep the length, then we substitute the width:

[tex]2(l+w)=100[/tex]

[tex]l+w = 50[/tex]

[tex]w=50-l[/tex]

[tex]l\times w\geq 400[/tex]

[tex]l(50-l)\geq 400[/tex]

[tex]50l-l^2\geq 400[/tex]

[tex]l^2-50l+400\leq 0[/tex]

[tex](l-40)(l-10)\leq 0[/tex]

Now, we can determine which intervals meet the condition by drawing the number line.

Based on the drawing, the interval which is less than or equal to 0 is 10 ≤ l ≤ 40. Therefore, the length should be longer than 10 feet but shorter than 40 feet.

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