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Charlene is knitting a baby blanket. She wants its width, [tex]\( w \)[/tex], to be at least half its length, [tex]\( l \)[/tex]. She estimates that she has enough yarn to put fringe around the blanket, as long as the perimeter of the blanket is no more than 180 inches. The system of inequalities shown represents the width of the blanket in inches, [tex]\( w \)[/tex], and the length in inches, [tex]\( l \)[/tex].

[tex]\[
\begin{array}{l}
w \geq 0.5l \\
2l + 2w \leq 180
\end{array}
\][/tex]

What is the maximum length possible for her blanket?

A. 30 inches
B. 45 inches
C. 60 inches
D. 90 inches



Answer :

GinnyAnswer
To determine the maximum length possible for Charlene's baby blanket, we need to consider the given conditions and solve the system of inequalities step-by-step.

The inequalities given are:
1. [tex]\( w \geq 0.5l \)[/tex] (The width must be at least half the length)
2. [tex]\( 2l + 2w \leq 180 \)[/tex] (The perimeter of the blanket must not exceed 180 inches)

Let's walk through the solution:

1. Simplify the Perimeter Inequality:
[tex]\[ 2l + 2w \leq 180 \][/tex]
We can simplify this by dividing every term by 2:
[tex]\[ l + w \leq 90 \][/tex]

2. Substitute the Width Constraint:
From the first inequality, we know:
[tex]\[ w \geq 0.5l \][/tex]

To find the maximum length [tex]\( l \)[/tex], we replace [tex]\( w \)[/tex] with the smallest possible value given by [tex]\( 0.5l \)[/tex]. So:
[tex]\[ l + 0.5l \leq 90 \][/tex]

3. Combine Like Terms:
Combine the terms on the left side:
[tex]\[ 1.5l \leq 90 \][/tex]

4. Solve for [tex]\( l \)[/tex]:
Divide both sides by 1.5 to isolate [tex]\( l \)[/tex]:
[tex]\[ l \leq 60 \][/tex]

This tells us that the maximum length [tex]\( l \)[/tex] can be is 60 inches.

To ensure this length satisfies all conditions, we calculate the corresponding width [tex]\( w \)[/tex]:
[tex]\[ w = 0.5 \times 60 \][/tex]
[tex]\[ w = 30 \][/tex]

Checking the perimeter:
[tex]\[ 2l + 2w = 2 \times 60 + 2 \times 30 = 120 + 60 = 180 \][/tex]
Thus, the perimeter condition is satisfied.

Therefore, the maximum possible length for Charlene's baby blanket is:
[tex]\[ l = 60 \text{ inches} \][/tex]

So, the correct answer is:
[tex]\[ \boxed{60 \text{ inches}} \][/tex]

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