Select the correct answer.
A rectangular sheet of steel is being cut so that the length is four times the width. The perimeter of the sheet must be less than 100 inches.
Which inequality can be used to find all possible lengths, $l$, of the steel sheet?
A. $\frac{5}{2} l>100$
B. $10 l<100$
C. $\frac{5}{2} l<100$
D. $10 l>100
Answer (2)
Define the length as l and the width as w , and express the width in terms of the length: w = 4 l .
Use the perimeter formula P = 2 l + 2 w and the given condition P < 100 .
Substitute w in the perimeter formula to get P = 2 5 l .
Form the inequality 2 5 l < 100 , which represents the constraint on the length. The final answer is 2 5 l < 100 .
Explanation
Understanding the Problem Let's analyze the problem. We are given that the length of a rectangular sheet of steel is four times its width, and the perimeter of the sheet must be less than 100 inches. We need to find an inequality that can be used to find all possible lengths, l , of the steel sheet.
Setting up the Equations Let l be the length and w be the width of the rectangular sheet. We are given that l = 4 w , which implies w = 4 l . The perimeter P of a rectangle is given by P = 2 l + 2 w . We are given that the perimeter must be less than 100 inches, so P < 100 .
Substituting and Simplifying Substitute w = 4 l into the perimeter formula: P = 2 l + 2 ( 4 l ) = 2 l + 2 l = 2 4 l + 2 l = 2 5 l . Since P < 100 , we have 2 5 l < 100 .
Finding the Inequality Therefore, the inequality that can be used to find all possible lengths l is 2 5 l < 100 .
Final Answer The correct answer is 2 5 l < 100 .
Examples
Imagine you're designing a rectangular garden where the length must be four times the width, and you have a limited amount of fencing (less than 100 inches) to enclose it. This problem helps you determine the maximum possible length of the garden while staying within your fencing limit. Understanding how to set up and solve inequalities in this context can guide you in real-world scenarios such as designing layouts, managing resources, or optimizing dimensions within constraints. This algebraic approach ensures you make the most of your available resources while adhering to specific requirements.
The inequality that represents the lengths of the rectangular steel sheet is 2 5 l < 100 . Therefore, the correct answer is option C. This means the calculated perimeter of the sheet must remain under 100 inches based on the defined dimensions.
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