Question

ernesto sews a rectangular blanket with a patterned center and a solid border. the patterned center has a length that is twice the width, w, in inches (in.). the solid border has a constant width of 16 inches, as shown. create an expression that represents the area of the solid border, in square inches.

Explanation:

Step1: Find dimensions of patterned center

The patterned center has width \( w \) and length \( 2w \). So area of center is \( A_{center} = 2w \times w = 2w^2 \).

Step2: Find dimensions of entire blanket

The border has width 16 in, so total length of blanket is \( 2w + 2\times16 = 2w + 32 \), total width is \( w + 2\times16 = w + 32 \). Area of blanket: \( A_{blanket}=(2w + 32)(w + 32) \).

Step3: Expand blanket area

\[

$$\begin{align*} (2w + 32)(w + 32)&=2w\times w + 2w\times32 + 32\times w + 32\times32\\ &=2w^2 + 64w + 32w + 1024\\ &=2w^2 + 96w + 1024 \end{align*}$$

\]

Step4: Subtract center area from blanket area

Area of border \( A_{border}=A_{blanket}-A_{center}=(2w^2 + 96w + 1024)-2w^2 = 96w + 1024 \). Alternatively, calculate border area by parts: two rectangles along length, two along width, and four corners. But subtraction is simpler.

Answer:

\( 96w + 1024 \) (or equivalent expanded form)