The perimeter of a rectangular pool is more than 62 meters, and the width is at least 10 meters less than the length. Which system of inequalities represents the possible length in meters, [tex]l[/tex], and the possible width in meters, [tex]w[/tex], of the pool?
A. [tex]
\begin{aligned}
w & \leq 10-1 \\
21+2 w & \geq 62
\end{aligned}
[/tex]
B. [tex]
w \leq 10-1
2 l+2 w\ \textgreater \ 62
[/tex]
C. [tex]
w \leq I-10
2 l+2 w \geq 62
[/tex]
D. [tex]
w \leq 1-10
2 l+2 w\ \textgreater \ 62
[/tex]
Answer (1)
Express the perimeter condition as an inequality: 62"> 2 l + 2 w > 62 .
Express the width condition as an inequality: $w
Combine the two inequalities to form the system of inequalities.
The correct system of inequalities is: 62 \end{cases}}"> { w ≤ l − 10 2 l + 2 w > 62 .
Explanation
Problem Analysis Let's analyze the given problem. We are given two conditions for a rectangular pool: its perimeter is more than 62 meters, and its width is at least 10 meters less than its length. We need to express these conditions as a system of inequalities involving the length l and the width w of the pool.
Perimeter Inequality The perimeter of a rectangle is given by 2 l + 2 w , where l is the length and w is the width. The problem states that the perimeter is more than 62 meters, so we can write this as: 62"> 2 l + 2 w > 62
Width Inequality The problem also states that the width is at least 10 meters less than the length. This can be written as: w This inequality means that the width $w$ is less than or equal to the length $l$ minus 10. 4. System of Inequalities Combining these two inequalities, we get the following system of inequalities: \begin{cases} 2l + 2w > 62 \ w \leq l - 10 \end{cases}
Final Answer Comparing this system of inequalities with the given options, we see that the correct option is:
62"> w 2 l + 2 w > 62
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, consider a delivery service optimizing routes. They need to ensure that the total distance traveled ( d ) is less than a certain limit (e.g., 100 miles) while also ensuring that the number of packages delivered ( p ) is more than a minimum requirement (e.g., 50 packages). This can be expressed as a system of inequalities: d < 100 and 50"> p > 50 . By solving such systems, the delivery service can efficiently plan routes and manage resources.
