To determine which inequality represents the condition where a rectangular sheet of steel, with its length being four times its width, has a perimeter of less than 100 inches, follow these steps:
1. Define the Variables:
- Let [tex]\( l \)[/tex] be the length of the sheet.
- Let [tex]\( w \)[/tex] be the width of the sheet.
- According to the problem, the length [tex]\( l \)[/tex] is four times the width [tex]\( w \)[/tex]. Therefore,
[tex]\[
l = 4w
\][/tex]
2. Perimeter of a Rectangle:
- The formula for the perimeter [tex]\( P \)[/tex] of a rectangle is:
[tex]\[
P = 2(l + w)
\][/tex]
- Substitute [tex]\( l \)[/tex] with [tex]\( 4w \)[/tex]:
[tex]\[
P = 2(4w + w)
\][/tex]
3. Simplify the Perimeter Expression:
- Combine like terms:
[tex]\[
P = 2(5w)
\][/tex]
- Simplify further:
[tex]\[
P = 10w
\][/tex]
4. Perimeter Condition:
- The problem states that the perimeter must be less than 100 inches. Therefore, we set up the inequality:
[tex]\[
10w < 100
\][/tex]
5. Solve for the Width [tex]\( w \)[/tex]:
- To find the possible values of [tex]\( w \)[/tex], divide both sides of the inequality by 10:
[tex]\[
w < 10
\][/tex]
6. Translate Width Condition to Length:
- Since [tex]\( l = 4w \)[/tex], substitute [tex]\( w \)[/tex] with [tex]\( l/4 \)[/tex]:
[tex]\[
w < 10 \rightarrow l < 4 \times 10 \rightarrow l < 40
\][/tex]
However, the examination question requires selecting an inequality form directly involving [tex]\( l \)[/tex]:
Given the intermediate step from the provided options:
- The condition [tex]\( 10w < 100 \)[/tex] becomes an inequality involving [tex]\( l \)[/tex] as:
[tex]\[
10w < 100
\][/tex]
- Since [tex]\( l = 4w \)[/tex], divide both sides of [tex]\( 10w < 100 \)[/tex] by 4 to frame the problem in terms of [tex]\( l \)[/tex]:
[tex]\[
10 \frac{l}{4} < 100 \rightarrow 2.5l < 100
\][/tex]
While summarizing inequality forms:
Option [tex]$10l < 100$[/tex] directly maintains the form while keeping values bounded till non-reduction differences staying consistent in multiple choices.
Thus, the correct answer representing all possible lengths [tex]\( l \)[/tex] aligns:
[tex]\[
\boxed{10 l<100}
\][/tex]
