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.
length $(l)$
width $(w)$
Which inequality can be used to find all possible lengths, $l$, of the steel sheet?
$\frac{1}{2} l>100$
$10 l>100$
$5l<100$
$10 l<100$
Answer (2)
Express the width in terms of the length: w = 4 l .
Substitute into the perimeter formula: P = 2 l + 2 ( 4 l ) = 2 5 l .
Set up the inequality: 2 5 l < 100 .
Simplify the inequality: 5 l < 200 . The closest answer among the options is 10 l < 100 , but it is not correct. The correct inequality derived from the problem statement is 5 l < 200 .
Explanation
Understanding the Problem Let's analyze the problem. We are given that the length l of a rectangular sheet is four times its width w , which means l = 4 w . The perimeter P of the sheet must be less than 100 inches, so P < 100 . We need to find an inequality that expresses this condition in terms of the length l .
Expressing Perimeter in Terms of Length The formula for the perimeter of a rectangle is P = 2 l + 2 w . Since l = 4 w , we can express the width in terms of the length as w = 4 l . Now, substitute this expression for w into the perimeter formula: P = 2 l + 2 w = 2 l + 2 ( 4 l ) Simplifying this, we get: P = 2 l + 2 l To combine these terms, we need a common denominator, which is 2: P = 2 4 l + 2 l = 2 5 l
Setting up the Inequality We know that the perimeter must be less than 100 inches, so we can write the inequality: 2 5 l < 100 To isolate l , we can multiply both sides of the inequality by 2: 5 l < 200
Finding the Correct Inequality Therefore, the inequality that can be used to find all possible lengths l of the steel sheet is 5 l < 200 . However, this is not one of the options. Let's divide both sides of the inequality 5 l < 200 by 5 to solve for l : l < 5 200 l < 40 Now, let's check the given options to see which one is equivalent to 2 5 l < 100 . Multiplying both sides of 2 5 l < 100 by 2 gives 5 l < 200 . The given option 10 l < 100 is incorrect. The correct inequality is 2 5 l < 100 , which simplifies to 5 l < 200 . However, the question asks for an inequality that can be used to find all possible lengths l . The inequality 5 l < 200 can be used to find all possible lengths. However, we need to find the correct option among the given choices. The correct option is 5 l < 100 is not present, but 10 l < 100 is not correct either. Let's re-examine the perimeter calculation. We have P = 2 l + 2 w and l = 4 w , so w = l /4 . Then P = 2 l + 2 ( l /4 ) = 2 l + l /2 = ( 4 l + l ) /2 = 5 l /2 . Since P < 100 , we have 5 l /2 < 100 . Multiplying by 2 gives 5 l < 200 . The question is incorrect. However, if the length was 10 times the width, then l = 10 w , so w = l /10 . Then P = 2 l + 2 w = 2 l + 2 ( l /10 ) = 2 l + l /5 = ( 10 l + l ) /5 = 11 l /5 . If 11 l /5 < 100 , then 11 l < 500 . The closest answer is 10 l < 100 , but that is incorrect. Let's go back to the original problem. We have l = 4 w and P < 100 . So P = 2 l + 2 w = 2 l + 2 ( l /4 ) = 2 l + l /2 = 5 l /2 < 100 . Multiplying by 2 gives 5 l < 200 . Dividing by 5 gives l < 40 . None of the options are correct. However, if we assume that the perimeter is P = 10 l , then 10 l < 100 .
Examples
Understanding perimeters and inequalities is crucial in various real-world scenarios. For instance, if you're designing a rectangular garden with a limited fence length, you need to determine the maximum possible dimensions while adhering to the perimeter constraint. Similarly, in manufacturing, engineers use these concepts to optimize the size of components within a device, ensuring they fit within specified boundaries while maximizing functionality. This problem demonstrates how mathematical inequalities help in practical design and optimization processes.
The appropriate inequality for the rectangular sheet's length, given its perimeter must be less than 100 inches and that the length is four times the width, is 5 l < 200 . However, among the multiple-choice options provided, none are correct. This means students should focus on deriving the correct inequalities based on the conditions given in the problem statement.
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