Advanced Placement (AP), 05.05.2020 23:11 mithydizon1553

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.

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Answer from: lifeisgaming37

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,

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

For the small cake, we have

a = 3 in.

h = 3 in.

Volume of small slice = $a^2 \times h \times \frac{\sqrt{3} }{4} = \frac{3^2\sqrt{3} }{4} \times 3 = \frac{27\sqrt{3} }{4} \ in^3$ .

For the big cake, we have

a = 4 in.

h = 4 in.

Volume of big slice = $a^2 \times h \times \frac{\sqrt{3} }{4} = \frac{4^2\sqrt{3} }{4} \times 4 = 16\sqrt{3} \ in^3$ .

Total volume of slice = Volume of small slice + Volume of big slice

Total volume of slice = $\frac{27\sqrt{3} }{4} \ in^3 +16\sqrt{3} \ in^3 = \frac{91\sqrt{3} }{4} \ in^3 = 39.404 \ in^3$

Total volume of slice = 39404 in³.

Answer from: Quest

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Answer from: Quest

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