Question

charlene is knitting a baby blanket. she wants its width, w, to be at least half its length, l. 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, w, and the length in inches, l.

( w geq 0.5l )
( 2l + 2w leq 180 )

what is the maximum length possible for her blanket?

• 30 inches
• 45 inches
• 60 inches
• 90 inches

Explanation:

Step1: Simplify the perimeter inequality

The perimeter inequality is \(2l + 2w\leq180\). Divide both sides by 2: \(l + w\leq90\), so \(w\leq90 - l\).

Step2: Substitute \(w\) from the first inequality

We know \(w\geq0.5l\) and \(w\leq90 - l\). For the maximum length, we consider the equality case (since we want to find the limit). So set \(w = 0.5l\) (the lower bound of \(w\) in terms of \(l\), which will help maximize \(l\) when substituted into the perimeter - related inequality). Substitute \(w = 0.5l\) into \(l+w\leq90\):
\(l + 0.5l\leq90\)
\(1.5l\leq90\)

Step3: Solve for \(l\)

Divide both sides of \(1.5l\leq90\) by \(1.5\):
\(l\leq\frac{90}{1.5}=60\)

Answer:

60 inches