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, [tex]$l$[/tex], and widths, [tex]$w$[/tex], 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 for the garden?
A. [tex]$l=20 ft ; w=5 ft$[/tex]
B. [tex]$l=20 ft ; w=10 ft$[/tex]
C. [tex]$l=60 ft ; w=20 ft$[/tex]
D. [tex]$I=55 ft ; w=30 ft$[/tex]
Answer (2)
Check if w ≥ 10 is satisfied.
Check if 2 l + 2 w ≤ 150 is satisfied.
For l = 20 and w = 5 , the first inequality is not satisfied.
For l = 20 and w = 10 , both inequalities are satisfied.
For l = 60 and w = 20 , the second inequality is not satisfied.
For l = 55 and w = 30 , the second inequality is not satisfied.
Possible dimensions for the garden are l = 20 f t ; w = 10 f t .
Explanation
Analyzing the Problem We are given a system of inequalities that models the possible dimensions of a rectangular garden. We need to check which of the given pairs of length l and width w satisfy both inequalities. The inequalities are:
w ≥ 10
2 l + 2 w ≤ 150
Let's analyze each option:
l = 20 f t ; w = 5 f t
l = 20 f t ; w = 10 f t
l = 60 f t ; w = 20 f t
l = 55 f t ; w = 30 f t
Checking Option 1 Option 1: l = 20 f t ; w = 5 f t
Check inequality 1: w ≥ 10 ⇒ 5 ≥ 10 . This is false.
Since the first inequality is not satisfied, we don't need to check the second inequality. This option is not possible.
Checking Option 2 Option 2: l = 20 f t ; w = 10 f t
Check inequality 1: w ≥ 10 ⇒ 10 ≥ 10 . This is true.
Check inequality 2: 2 l + 2 w ≤ 150 ⇒ 2 ( 20 ) + 2 ( 10 ) ≤ 150 ⇒ 40 + 20 ≤ 150 ⇒ 60 ≤ 150 . This is true.
Since both inequalities are satisfied, this option is possible.
Checking Option 3 Option 3: l = 60 f t ; w = 20 f t
Check inequality 1: w ≥ 10 ⇒ 20 ≥ 10 . This is true.
Check inequality 2: 2 l + 2 w ≤ 150 ⇒ 2 ( 60 ) + 2 ( 20 ) ≤ 150 ⇒ 120 + 40 ≤ 150 ⇒ 160 ≤ 150 . This is false.
Since the second inequality is not satisfied, this option is not possible.
Checking Option 4 Option 4: l = 55 f t ; w = 30 f t
Check inequality 1: w ≥ 10 ⇒ 30 ≥ 10 . This is true.
Check inequality 2: 2 l + 2 w ≤ 150 ⇒ 2 ( 55 ) + 2 ( 30 ) ≤ 150 ⇒ 110 + 60 ≤ 150 ⇒ 170 ≤ 150 . This is false.
Since the second inequality is not satisfied, this option is not possible.
Final Answer Only the dimensions l = 20 f t and w = 10 f t satisfy both inequalities. Therefore, these are possible dimensions for the garden.
Examples
Understanding constraints and inequalities is crucial in resource allocation. For instance, a farmer might have a limited amount of land and water and needs to decide how much of each crop to plant to maximize profit. Similarly, a manufacturer might have constraints on production time and materials and needs to optimize the production schedule. These problems can be modeled using systems of inequalities, allowing for informed decision-making.
The only dimensions that satisfy the conditions for Irina's garden are l = 20 f t and w = 10 f t . Therefore, the correct option is B. The analysis showed that all other options failed to meet the specified inequalities.
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