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.
[tex]
\begin{aligned}
w & \geq 10 \\
2 l+2 w & \leq 150
\end{aligned}
[/tex]
Which length and width are possible dimensions of the garden?
A. $l=20 ft ; w=5 ft$
B. $l=20 ft ; w=10 ft$
C. $l=60 ft ; w=20 ft$
D. $l=55 ft ; w=30 ft$
Answer (1)
Check each pair of length and width against the given inequalities.
Pair 1 ( l = 20 ft, w = 5 ft) fails because w must be at least 10 ft.
Pair 2 ( l = 20 ft, w = 10 ft) satisfies both inequalities.
Pairs 3 and 4 fail because their perimeters exceed 150 ft.
The only possible dimensions are l = 20 f t ; w = 10 f t .
Explanation
Understanding the Problem We are given a system of inequalities that models the possible lengths, l , and widths, w , of a rectangular garden:
w ≥ 10
2 l + 2 w ≤ 150
We need to determine which of the given length and width pairs satisfy both inequalities.
Testing Pair 1 Let's test each pair:
Pair 1: l = 20 ft, w = 5 ft
Check w ≥ 10 : 5 ≥ 10 is false. Since the first inequality is not satisfied, we don't need to check the second inequality.
Testing Pair 2 Pair 2: l = 20 ft, w = 10 ft
Check w ≥ 10 : 10 ≥ 10 is true. Check 2 l + 2 w ≤ 150 : 2 ( 20 ) + 2 ( 10 ) ≤ 150 ⇒ 40 + 20 ≤ 150 ⇒ 60 ≤ 150 is true. Since both inequalities are satisfied, this is a possible solution.
Testing Pair 3 Pair 3: l = 60 ft, w = 20 ft
Check w ≥ 10 : 20 ≥ 10 is true. Check 2 l + 2 w ≤ 150 : 2 ( 60 ) + 2 ( 20 ) ≤ 150 ⇒ 120 + 40 ≤ 150 ⇒ 160 ≤ 150 is false. Since the second inequality is not satisfied, this is not a possible solution.
Testing Pair 4 Pair 4: l = 55 ft, w = 30 ft
Check w ≥ 10 : 30 ≥ 10 is true. Check 2 l + 2 w ≤ 150 : 2 ( 55 ) + 2 ( 30 ) ≤ 150 ⇒ 110 + 60 ≤ 150 ⇒ 170 ≤ 150 is false. Since the second inequality is not satisfied, this is not a possible solution.
Final Answer Only Pair 2 satisfies both inequalities. Therefore, the possible dimensions for the garden are l = 20 ft and w = 10 ft.
Examples
Understanding systems of inequalities is crucial in various real-world scenarios, such as optimizing resource allocation under constraints. For instance, a farmer might use inequalities to determine the optimal mix of crops to plant, given limitations on land, water, and fertilizer. Similarly, a manufacturer could use inequalities to maximize production output while adhering to constraints on labor, materials, and budget. These mathematical tools enable informed decision-making and efficient resource management in diverse fields.
