Question

A wall with an area of (3x2 + 4x + 1) square units has a rectangular window that has a width of (x + 2) units and a length of (x + 1) units. The wall also has a built- in shelving unit that occupies an area of (x2 + 5x + 6) square units of wall space. If the wall is to be covered with wallpaper, how much wallpaper will be required?

Answer 1

x^2 - 4x - 7

Explanation:

Given:-

- The available total area of the wall, A_p = 3x2 + 4x + 1 units^2

- The dimensions of window, A_w = ( x + 2 ) x ( x + 1 ) units^2

- The dimensions of the shelving unit, A_s = x2 + 5x + 6 units^2

Find:-

If the wall is to be covered with wallpaper, how much wallpaper will be required?

Solution:-

- First we need to realize that the amount wallpaper used can be classified as the coverage area of the wall.

- The wall has a window and shelving unit that will discount the use of wallpaper for these regions. So we can subtracts the area of wall and shelving unit from the area of wall to determine the required wallpaper.

A_paper = A_p - A_w - A_s

A_paper = (3x^2 + 4x + 1) - [ ( x + 2 )*( x + 1 ) ] - (x^2 + 5x + 6)

A_paper = (3x^2 + 4x + 1) - (x^2 + 3x + 2) - (x^2 + 5x + 6)

- Simplify:

A_paper = (x^2 - 4x - 7)

- The amount of paper required to cover the wall is : x^2 - 4x - 7

Answer 2

The question seems to ask available area and not how much wallpaper is required. Let's calculate the area of the rectangular window. Area of a rectangle = Length * Breath Area = (x + 2) * (x + 1) = x^2 + 3x + 2 The wall also has a built- in shelving unit that occupies an area of (x^2 + 5x + 6). So total area occupied by the rectangular wall and shelf is (x^2 + 3x + 2) + (x^2 + 5x + 6) = 2x^2 + 8x + 8. If the wall is to be covered with wallpaper, the total available area = (3x^2 + 4x + 1) - (2x^2 + 8x + 8) = x^2 - 4x - 7