Six square windows of side length 2 m are to be placed into 12 m wide by 8.5 m high wall as shown. The windows are to be positioned so that the vertical spacing between the windows and the wall edges are equal. Similarly, the horizontal spacings are also equal.
a i) Find the horizontal distance between the windows.
ii) Find the vertical distance between the windows.
b) Find the area of the wall not including the window spaces.
c) If the wall included 3 rows of 4 windows (instead of 2 rows of 3) investigate if it would be possible to space all the windows so that the horizontal and vertical spacings are uniform (although not necessarily equal to each other).
The width is 12m. The 3 windows in a row require 6m
That leaves 6m to divide into 4 equal parts of 1.5m each
The height is 8.5m. The two windows in each column use up 4m
That leaves 4.5m to divide into 3 equal parts of 1.5m each
I think you can probably finish it up now. If you get stuck, come back and show us what you have done.
a i) Sure, I'd be happy to help! To find the horizontal distance between the windows, you can start by dividing the width of the wall by the number of spaces between the windows. In this case, there are 3 spaces between the 2 windows. So, the horizontal distance between the windows would be:
Horizontal distance = (Wall width - Sum of window widths) / Number of spaces between windows
Horizontal distance = (12 m - (2 m + 2 m)) / 3
Horizontal distance = (12 m - 4 m) / 3
Horizontal distance = 8 m / 3
Horizontal distance ≈ 2.67 m
ii) Similarly, to find the vertical distance between the windows, you can divide the height of the wall by the number of spaces between the windows. Here, there is 1 space between the 2 windows, so the vertical distance between the windows would be:
Vertical distance = (Wall height - Sum of window heights) / Number of spaces between windows
Vertical distance = (8.5 m - (2 m)) / 1
Vertical distance = 6.5 m / 1
Vertical distance = 6.5 m
b) To find the area of the wall not including the window spaces, you can find the total area of the wall and subtract the area of the windows. The total area of the wall is the product of its width and height:
Total area of wall = Wall width * Wall height
Total area of wall = 12 m * 8.5 m
Total area of wall = 102 m²
Since each window has an area of 2 m * 2 m = 4 m², and there are 6 windows, the area of all the windows is:
Total area of windows = Number of windows * Window area
Total area of windows = 6 * 4 m²
Total area of windows = 24 m²
Area of wall not including window spaces = Total area of wall - Total area of windows
Area of wall not including window spaces = 102 m² - 24 m²
Area of wall not including window spaces = 78 m²
c) If the wall included 3 rows of 4 windows, it may not be possible to space all the windows so that the horizontal and vertical spacings are uniform. Depending on the size of the windows and the dimensions of the wall, there might not be enough space to maintain equal spacings in both directions. It would require further investigation to determine the feasibility.
a)
i) To find the horizontal distance between the windows, we need to determine the total width occupied by the windows and calculate the space left in between.
Each window has a side length of 2 m, so its width is 2 m. In each row, we have three windows, so the total width occupied by the windows in a row is: 3 windows x 2 m = 6 m.
The wall is 12 m wide, and we have two rows of windows. Therefore, the total width occupied by the windows is: 2 rows x 6 m = 12 m.
The space left in between the windows is the difference between the total wall width and the width occupied by the windows: 12 m - 12 m = 0 m.
Therefore, the horizontal distance between the windows is 0 meters.
ii) Similarly, to find the vertical distance between the windows, we need to determine the total height occupied by the windows and calculate the space left in between.
Each window has a side length of 2 m, so its height is also 2 m. In each column, we have two windows, so the total height occupied by the windows in a column is: 2 windows x 2 m = 4 m.
The wall is 8.5 m high, and we have two columns of windows. Therefore, the total height occupied by the windows is: 2 columns x 4 m = 8 m.
The space left in between the windows is the difference between the total wall height and the height occupied by the windows: 8.5 m - 8 m = 0.5 m.
Therefore, the vertical distance between the windows is 0.5 meters.
b) The area of the wall not including the window spaces can be calculated by subtracting the area occupied by the windows from the total area of the wall.
The total area of the wall is: 12 m x 8.5 m = 102 m^2.
The area occupied by the windows is: 6 m (width occupied by the windows) x 8 m (height occupied by the windows) = 48 m^2.
Therefore, the area of the wall not including the window spaces is: 102 m^2 - 48 m^2 = 54 m^2.
c) If the wall included 3 rows of 4 windows instead of 2 rows of 3, it would still be possible to space all the windows so that the horizontal and vertical spacings are uniform.
The horizontal spacing between the windows would be the same as in part a) i), which is 0 meters.
The vertical spacing between the windows would also remain the same as in part a) ii), which is 0.5 meters.
Therefore, it is possible to space all the windows so that the horizontal and vertical spacings are uniform.
To find the horizontal and vertical distances between the windows, we need to consider the dimensions of the wall and the windows. Let's begin by solving part a) of the question.
a)
i) Horizontal distance between the windows:
Since there are 2 rows of 3 windows, the total width occupied by the windows is 3 windows multiplied by their side length, which gives us 3 * 2m = 6m. The remaining wall space on either side of the windows is 12m - 6m = 6m. To find the horizontal distance between the windows, we divide this remaining space equally between the windows. The distance between each window is therefore 6m / (3 + 1) = 6m / 4 = 1.5m.
ii) Vertical distance between the windows:
Similarly, we can find the vertical distance between the windows. The total height occupied by the windows is 2 windows multiplied by their side length, which gives us 2 * 2m = 4m. The remaining wall space above and below the windows is 8.5m - 4m = 4.5m. To find the vertical distance between the windows, we divide this remaining space equally between the windows. The distance between each window is therefore 4.5m / (2 + 1) = 4.5m / 3 = 1.5m.
b)
To find the area of the wall not including the window spaces, we need to subtract the total area occupied by the windows from the total area of the wall.
The total area of the wall is the product of its width and height, which is 12m * 8.5m = 102m².
Since each window has an area of 2m * 2m = 4m², the total area occupied by the windows is 3 windows * 2 rows * 4m² = 24m².
Therefore, the area of the wall not including the window spaces is 102m² - 24m² = 78m².
c)
If the wall included 3 rows of 4 windows instead of 2 rows of 3, it would still be possible to space all the windows so that the horizontal and vertical spacings are uniform, although not necessarily equal to each other. The process would be similar to part a), but with different dimensions and calculations based on the new arrangement of the windows.