A cake has two layers. Each layer is a regular hexagonal prism. You cut and remove a slice that takes away one face of each prism as shown. What is the volume of the slice? What is the volume of the remaining cake? Use pencil and paper. Describe two ways to find the volume of the slice.
Answers
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 = \frac{1}{2} \times Base \times Height = \frac{1}{2} \times 3 \times 3\times sin(60) = \frac{1}{2} \times 3 \times 3 \times \frac{\sqrt{3} }{2} = \frac{9\sqrt{3} }{4}21×Base×Height=21×3×3×sin(60)=21×3×3×23=493
Volume of small slice = Area of surface small slice × Height of small cake
= \frac{9\sqrt{3} }{4} \times 3 = \frac{27\sqrt{3} }{4} =11.69 \ in^3 \approx 12 \ in^3493×3=4273=11.69 in3≈12 in3
For the big cake, we have;
Big cake slice triangle side = 4 in.
Area of surface of big slice = \frac{1}{2} \times Base \times Height = \frac{1}{2} \times 4 \times 4\times sin(60) = 8 \times \frac{\sqrt{3} }{2} = 4\sqrt{3}21×Base×Height=21×4×4×sin(60)=8×23=43
Volume of big slice = Area of surface of big slice × Height of big slice
= 4\sqrt{3} \times 4 = 16\sqrt{3} =27.71 \ in^3 \approx 28 \ in^343×4=163=27.71 in3≈28 in3
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 = 5 \times \frac{27\sqrt{3} }{4} = 5 \times 11.69 \ in^3 = 58.45 \ in^35×4273=5×11.69 in3=58.45 in3
For the big cake the remaining volume = 5 \times 16\sqrt{3} = 5 \times 27.71 \ in^3 = 138.56 \ in^35×163=5×27.71 in3=138.56 in3
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;
Volume\, of \, regular \, hexagon, \ A = \frac{3\sqrt{3} }{2} a^2 \times hVolumeofregularhexagon, A=233a2×h
Where:
a = Length of side
h = Height of hexagon
The volume of each slice is therefore,
\frac{Volume\, of \, regular \, hexagon, \ A }{6} =\frac{ \frac{3\sqrt{3} }{2} a^2 \times h}{6} = a^2 \times h \times \frac{3\sqrt{3} }{12} = a^2 \times h \times \frac{\sqrt{3} }{4}6Volumeofregularhexagon, A=6233a2×h=a2×h×1233=a2×h×43
For the small cake, we have
a = 3 in.
h = 3 in.
Volume of small slice = a^2 \times h \times \frac{\sqrt{3} }{4} = \frac{3^2\sqrt{3} }{4} \times 3 = \frac{27\sqrt{3} }{4} \ in^3a2×h×43=43