Question

lila is making a large blanket that needs to cover at least 132 square feet. the blanket needs to be one foot longer than it is wide. what do the dimensions (length and width) of her blanket need to be? prove your answer in more than one way.
step 1: write the inequality in standard form.
(make sure one side is zero.)
step 2: factor the quadratic expression.
(break it down into two binomials.)
step 3: find the roots.
(set each factor equal to zero to find
the boundary points.)
step 4: test the intervals
(use the roots to divide the number line
into intervals and test one value from each.)
step 5: write the solutions set.

Explanation:

Step 1: Define variables and inequality

Let width be \( w \) feet, length is \( w + 1 \) feet. Area \( A=w(w + 1)\geq132 \). Rewrite as \( w^{2}+w - 132\geq0 \).

Step 2: Factor the quadratic

Find two numbers: product \(-132\), sum \(1\). Numbers \(12\) and \(-11\). So \( w^{2}+w - 132=(w + 12)(w - 11) \).

Step 3: Find the roots

Set \((w + 12)(w - 11)=0\). Roots: \( w=-12 \) (discard, width can't be negative), \( w = 11 \).

Step 4: Test intervals

Interval 1: \( w<11 \) (e.g., \( w = 0 \)): \((0 + 12)(0 - 11)=-132<0\), doesn't satisfy \( \geq0 \).
Interval 2: \( w>11 \) (e.g., \( w = 12 \)): \((12 + 12)(12 - 11)=24>0\), satisfies.
Boundary point \( w = 11 \): \( 11(11 + 1)=132 \), satisfies.

Step 5: Solution set

Width \( w\geq11 \) feet, length \( l=w + 1\geq12 \) feet.

Another way: Solve \( w^{2}+w - 132 = 0 \) using quadratic formula \( w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \), \( a = 1 \), \( b = 1 \), \( c=-132 \). \( w=\frac{-1\pm\sqrt{1 + 528}}{2}=\frac{-1\pm\sqrt{529}}{2}=\frac{-1\pm23}{2} \). Positive root \( \frac{22}{2}=11 \), so \( w\geq11 \), \( l\geq12 \).

Answer:

Width of the blanket should be at least \( 11 \) feet and length at least \( 12 \) feet (since length \(=\) width \( + 1\), and area \(=\) width \(\times\) length \(\geq132\) square feet, verified by factoring and quadratic formula).