Question

irina wants to build a fence around a rectangular vegetable garden so that it has a width of at least 10 feet. she can use a maximum of 150 feet of fencing. the system of inequalities that models the possible lengths, \\(l\\), and widths, \\(w\\), of her garden is shown.

\\(w \ge 10\\)
\\(2l + 2w \le 150\\)

which length and width are possible dimensions for the garden?

• \\(l = 20\text{ ft}; w = 10\text{ ft}\\)
• \\(l = 20\text{ ft}; w = 5\text{ ft}\\)
• \\(l = 60\text{ ft}; w = 20\text{ ft}\\)
• \\(l = 55\text{ ft}; w = 30\text{ ft}\\)

Explanation:

Identify the system of inequalities

The problem provides a system of inequalities modeling the dimensions of a rectangular garden:

1. \(w \ge 10\)
2. \(2l + 2w \le 150\)

Test the first option

For \(l = 20\text{ ft}\) and \(w = 10\text{ ft}\):

• Check \(w \ge 10\): \(10 \ge 10\) (True)
• Check \(2l + 2w \le 150\): \(2(20) + 2(10) = 40 + 20 = 60 \le 150\) (True)

Test the second option

For \(l = 20\text{ ft}\) and \(w = 5\text{ ft}\):

• Check \(w \ge 10\): \(5 \ge 10\) (False)

Test the third option

For \(l = 60\text{ ft}\) and \(w = 20\text{ ft}\):

• Check \(w \ge 10\): \(20 \ge 10\) (True)
• Check \(2l + 2w \le 150\): \(2(60) + 2(20) = 120 + 40 = 160 \le 150\) (False)

Test the fourth option

For \(l = 55\text{ ft}\) and \(w = 30\text{ ft}\):

• Check \(w \ge 10\): \(30 \ge 10\) (True)
• Check \(2l + 2w \le 150\): \(2(55) + 2(30) = 110 + 60 = 170 \le 150\) (False)

Answer:

• (A) \(l = 20\text{ ft}; w = 10\text{ ft}\) (Correct answer)
• (B) \(l = 20\text{ ft}; w = 5\text{ ft}\)
• (C) \(l = 60\text{ ft}; w = 20\text{ ft}\)
• (D) \(l = 55\text{ ft}; w = 30\text{ ft}\)