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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 pris

Ivan

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 = $\frac{1}{2} \times Base \times Height = \frac{1}{2} \times 3 \times 3\times sin(60) = \frac{1}{2} \times 3 \times 3 \times \frac{\sqrt{3} }{2} = \frac{9\sqrt{3} }{4}$

Volume of small slice = Area of surface small slice × Height of small cake

= $\frac{9\sqrt{3} }{4} \times 3 = \frac{27\sqrt{3} }{4} =11.69 \ in^3 \approx 12 \ in^3$

For the big cake, we have;

Big cake slice triangle side = 4 in.

Area of surface of big slice = $\frac{1}{2} \times Base \times Height = \frac{1}{2} \times 4 \times 4\times sin(60) = 8 \times \frac{\sqrt{3} }{2} = 4\sqrt{3}$

Volume of big slice = Area of surface of big slice × Height of big slice

= $4\sqrt{3} \times 4 = 16\sqrt{3} =27.71 \ in^3 \approx 28 \ in^3$

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 = $5 \times \frac{27\sqrt{3} }{4} = 5 \times 11.69 \ in^3 = 58.45 \ in^3$

For the big cake the remaining volume = $5 \times 16\sqrt{3} = 5 \times 27.71 \ in^3 = 138.56 \ in^3$

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;

$Volume\, of \, regular \, hexagon, \ A = \frac{3\sqrt{3} }{2} a^2 \times h$

Where:

a = Length of side

h = Height of hexagon

The volume of each slice is therefore,

$\frac{Volume\, of \, regular \, hexagon, \ A }{6} =\frac{ \frac{3\sqrt{3} }{2} a^2 \times h}{6} = a^2 \times h \times \frac{3\sqrt{3} }{12} = a^2 \times h \times \frac{\sqrt{3} }{4}$