Question

activity 3: a graphic designer is developing images for icons. the square pixelated image is placed inside a border that is 2 pixels wide on all sides. if the area of the border of the image is 176 square pixels, what is the area of the image?

Explanation:

Step1: Find the area of the outer - square and inner - square

The side length of the outer - square is $x + 4$ and the side length of the inner - square is $x$. The area of the outer - square is $(x + 4)^2$ and the area of the inner - square is $x^2$. The area of the border is the difference between the area of the outer - square and the area of the inner - square, so $(x + 4)^2-x^2=176$.

Step2: Expand the left - hand side of the equation

Expand $(x + 4)^2$ using the formula $(a + b)^2=a^{2}+2ab + b^{2}$, where $a=x$ and $b = 4$. So $(x + 4)^2=x^{2}+8x + 16$. Then the equation becomes $x^{2}+8x + 16-x^{2}=176$.

Step3: Simplify the equation

The $x^{2}$ terms cancel out, leaving $8x+16 = 176$.

Step4: Solve for $x$

Subtract 16 from both sides of the equation: $8x=176 - 16=160$. Then divide both sides by 8, so $x = 20$.

Step5: Find the area of the image (inner - square)

The area of the inner - square (the image) is $x^2$. Substitute $x = 20$ into it, and the area is $20^2=400$ square pixels.

Answer:

400 square pixels