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]$l=55 ft ; w=30 ft$[/tex]
Answer (1)
Check if w ≥ 10 and 2 l + 2 w ≤ 150 for each pair.
For l = 20 , w = 5 : 5 ≥ 10 is false.
For l = 20 , w = 10 : 10 ≥ 10 is true, 2 ( 20 ) + 2 ( 10 ) = 40 + 20 = 60 ≤ 150 is true.
For l = 60 , w = 20 : 20 ≥ 10 is true, 2 ( 60 ) + 2 ( 20 ) = 120 + 40 = 160 ≤ 150 is false.
For l = 55 , w = 30 : 30 ≥ 10 is true, 2 ( 55 ) + 2 ( 30 ) = 110 + 60 = 170 ≤ 150 is false.
Therefore, only l = 20 ft and w = 10 ft are possible dimensions. I = 20 f t ; w = 10 f t
Explanation
Checking the inequalities We need to check each pair of length l and width w to see if they satisfy both inequalities:
w ≥ 10
2 l + 2 w ≤ 150
Examples
Understanding inequalities helps in resource allocation. For example, if you have a budget and need to buy certain items, inequalities can help you determine the possible quantities of each item you can afford. This is useful in personal finance, business planning, and many other areas where constraints exist.
