A wall of a building is made from blocks that are each 1 cubic foot. The wall is 6 feet high and 6 feet wide. A window is made by removing some blocks as shown below. The window is 2 feet high by 2 feet wide.
The wall is 6 feet high and 6 feet wide. The window is 2 feet high by 2 feet wide and is cut from the center of the wall.
Suppose the wall is expanded to be 12 feet high by 12 feet wide, and the window is expanded to be 4 feet high by 4 feet wide. How will this change the volume of the wall?
A.
The volume will not change.
B.
The volume of the wall will double.
C.
The volume of the wall will increase by 96 cubic feet.
D.
The volume of the wall will increase by 128 cubic feet.
A. The volume will not change. The expansion of the wall and window is proportional, so the ratio of wall volume to window volume remains the same. Therefore, the added volume from expanding the wall is exactly balanced by the volume removed in expanding the window.
The original volume of the wall can be calculated by multiplying its height (6 feet), width (6 feet), and depth (1 foot). So, the volume is 6 feet * 6 feet * 1 foot = 36 cubic feet.
The original volume of the window can be calculated by multiplying its height (2 feet), width (2 feet), and depth (1 foot). So, the volume is 2 feet * 2 feet * 1 foot = 4 cubic feet.
To find the remaining volume of the wall after window removal, we subtract the volume of the window from the original volume of the wall: 36 cubic feet - 4 cubic feet = 32 cubic feet.
Now, let's calculate the new volume of the wall after expansion. The new height of the wall is 12 feet, the new width is 12 feet, and the depth remains the same at 1 foot. So, the new volume is 12 feet * 12 feet * 1 foot = 144 cubic feet.
To find the change in volume, we subtract the original remaining volume of the wall (32 cubic feet) from the new volume of the wall (144 cubic feet): 144 cubic feet - 32 cubic feet = 112 cubic feet.
Therefore, the correct answer is:
D. The volume of the wall will increase by 112 cubic feet.
To find the volume of the wall before and after the expansion, we need to multiply the height, width, and depth of the wall.
Before the expansion:
The height of the wall is 6 feet.
The width of the wall is 6 feet.
The depth of the wall is 1 foot (since each block is 1 cubic foot).
So, the volume of the wall before the expansion is 6 feet * 6 feet * 1 foot = 36 cubic feet.
After the expansion:
The height of the wall after the expansion is 12 feet.
The width of the wall after the expansion is 12 feet.
The depth of the wall is still 1 foot.
So, the volume of the wall after the expansion is 12 feet * 12 feet * 1 foot = 144 cubic feet.
The volume of the window before and after the expansion can be calculated in the same way, since it is also a rectangular shape.
Before the expansion:
The height of the window is 2 feet.
The width of the window is 2 feet.
The depth of the window is 1 foot.
So, the volume of the window before the expansion is 2 feet * 2 feet * 1 foot = 4 cubic feet.
After the expansion:
The height of the window after the expansion is 4 feet.
The width of the window after the expansion is 4 feet.
The depth of the window is still 1 foot.
So, the volume of the window after the expansion is 4 feet * 4 feet * 1 foot = 16 cubic feet.
To find the change in volume, we subtract the initial volume from the final volume:
Change in volume = volume after expansion - volume before expansion
Change in volume = 144 cubic feet - 36 cubic feet
Change in volume = 108 cubic feet.
Therefore, the volume of the wall will increase by 108 cubic feet.
None of the provided answer choices match our result of 108 cubic feet. It seems there is an error in the answer choices provided.