Question

Digital picture

Suppose that a digital picture, including its borders is x + 2 pixels by x +2 pixels and that the actual picture inside the border is  x -2 pixels by x – 2 pixels.

 

1. Find a polynomial that gives the number of pixels in the border.
2. Evaluate the polynomial in part (a) for x=5.
3. Sketch a digital picture of the letter H with x=5. Does the picture agree with the answer in part (b)?
4. Digital pictures typically have large values for x. if a picture has x=500 find the total number of pixels in this border.

Answer 1

(a) So the number of pixels in the border is:

(x + 2)(x + 2) - (x - 2)(x - 2) = x^2 + 4x + 4 - (x^2 - 4x + 4)

= 8x

(b) So when x = 5, (a) = 8 x 5 = 40

(c) It would be best to do this on some kind of graph paper: Draw a square which is 7 cm long and 7 cm wide (obviously it doesn't matter what size the squares are, I’m just using cm for convenience), then inside this box draw another box that is 3 cm long and 3 cm wide, then count how many square cm there are in the border area (the area between the big box and the small box). This will come to 40 pixels, just as in part (b)

(d) Simply place the value x = 500 into the original equation that you worked out, i.e.

no. of pixels = 8x = 8 x 500 = 4000