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Answer:
Volume of slice is approximately 40 in³
Volume of the remaining cake is 197.014 in³
Explanation:
Here we have two regular hexagons
one top small hexagon cake with side length = 3 in, height = 3 in
One big hexagon cake, side length = 4 in, Height = 4 in
A slice cut such the it removes a side segment is equivalent to an equilateral triangle with side length = length of hexagon side
Also all angles within the equilateral triangle are 60° each
Therefore, the length of the side of the removed equilateral triangle side is given as follows;
Top small cake slice triangle side = 3 in.
Area of surface of small slice =
Volume of small slice = Area of surface small slice × Height of small cake
=
For the big cake, we have;
Big cake slice triangle side = 4 in.
Area of surface of big slice =
Volume of big slice = Area of surface of big slice × Height of big slice
=
Total volume of slice = Volume of small slice + Volume of big slice
Total volume of slice = 12 in³ +28 in³ = 40 in³
The volume of the remaining cake can be found by noting that there were 6 possible slices of cake based on the 6 sides of the hexagon, since we removed 1 slice, the remaining 5 slices will have a volume given by multiplying the volume of 1 slice by 5 as follows;
For the small cake, the remaining volume =
For the big cake the remaining volume =
Total volume remaining cake = 58.45 in³ + 138.56 in³ = 197.014 in³
Together with the above way to find the volume of slice of cake, the volume of the slice can also be found by considering that the cake, with a shape of a regular hexagon is made up of 6 such slices. Therefore, if the volume of a regular hexagon is as follows;
Where:
a = Length of side
h = Height of hexagon
The volume of each slice is therefore,
For the small cake, we have
a = 3 in.
h = 3 in.
Volume of small slice = .
For the big cake, we have
a = 4 in.
h = 4 in.
Volume of big slice = .
Total volume of slice = Volume of small slice + Volume of big slice
Total volume of slice =
Total volume of slice = 39404 in³.