A wants to build a fence around a rectangular vegetable garden so that it has a width of at least 10 feet. A can use a maximum of 150 feet of fencing. The system of inequalities that models the possible lengths, $l$, widths, $w$, of the garden is shown.
$\begin{aligned}
w & \geq 10 \\
2 l+2 w & \leq 150
\end{aligned}$
Which length and width are possible dimensions for 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 if the width w is greater than or equal to 10.
Check if 2 l + 2 w is less than or equal to 150.
For l = 20 ft and w = 10 ft, both conditions are met: 10 ≥ 10 and 2 ( 20 ) + 2 ( 10 ) = 60 ≤ 150 .
Therefore, the possible dimensions are l = 20 f t ; w = 10 f t .
Explanation
Understanding the Problem We are given two inequalities that model the possible dimensions of a rectangular garden: w ≥ 10 and 2 l + 2 w ≤ 150 , where l is the length and w is the width. We need to check which of the given pairs ( l , w ) satisfy both inequalities.
Checking the First Pair Let's check the first pair: l = 20 ft, w = 5 ft. First, we check if w ≥ 10 . Since 5 < 10 , this pair does not satisfy the first inequality.
Checking the Second Pair Next, let's check the second pair: l = 20 ft, w = 10 ft. First, we check if w ≥ 10 . Since 10 ≥ 10 , this inequality is satisfied. Now, we check if 2 l + 2 w ≤ 150 . Substituting the values, we get 2 ( 20 ) + 2 ( 10 ) = 40 + 20 = 60 . Since 60 ≤ 150 , this inequality is also satisfied. Therefore, this pair is a possible solution.
Checking the Third Pair Now, let's check the third pair: l = 60 ft, w = 20 ft. First, we check if w ≥ 10 . Since 20 ≥ 10 , this inequality is satisfied. Now, we check if 2 l + 2 w ≤ 150 . Substituting the values, we get 2 ( 60 ) + 2 ( 20 ) = 120 + 40 = 160 . Since 150"> 160 > 150 , this inequality is not satisfied. Therefore, this pair is not a possible solution.
Checking the Fourth Pair Finally, let's check the fourth pair: l = 55 ft, w = 30 ft. First, we check if w ≥ 10 . Since 30 ≥ 10 , this inequality is satisfied. Now, we check if 2 l + 2 w ≤ 150 . Substituting the values, we get 2 ( 55 ) + 2 ( 30 ) = 110 + 60 = 170 . Since 150"> 170 > 150 , this inequality is not satisfied. Therefore, this pair is not a possible solution.
Final Answer Only the second pair, l = 20 ft and w = 10 ft, satisfies both inequalities.
Examples
Understanding constraints and inequalities is crucial in resource management. For instance, a farmer might want to maximize crop yield within limited resources like land and water. By setting up inequalities that represent these constraints, the farmer can determine the optimal planting area for each crop to achieve the highest possible yield. This method ensures efficient use of resources and maximizes productivity.
