Respuesta :
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 = [tex]\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}[/tex]
Volume of small slice = Area of surface small slice × Height of small cake
= [tex]\frac{9\sqrt{3} }{4} \times 3 = \frac{27\sqrt{3} }{4} =11.69 \ in^3 \approx 12 \ in^3[/tex]
For the big cake, we have;
Big cake slice triangle side = 4 in.
Area of surface of big slice = [tex]\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}[/tex]
Volume of big slice = Area of surface of big slice × Height of big slice
= [tex]4\sqrt{3} \times 4 = 16\sqrt{3} =27.71 \ in^3 \approx 28 \ in^3[/tex]
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 = [tex]5 \times \frac{27\sqrt{3} }{4} = 5 \times 11.69 \ in^3 = 58.45 \ in^3[/tex]
For the big cake the remaining volume = [tex]5 \times 16\sqrt{3} = 5 \times 27.71 \ in^3 = 138.56 \ in^3[/tex]
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;
[tex]Volume\, of \, regular \, hexagon, \ A = \frac{3\sqrt{3} }{2} a^2 \times h[/tex]
Where:
a = Length of side
h = Height of hexagon
The volume of each slice is therefore,
[tex]\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}[/tex]
For the small cake, we have
a = 3 in.
h = 3 in.
Volume of small slice = [tex]a^2 \times h \times \frac{\sqrt{3} }{4} = \frac{3^2\sqrt{3} }{4} \times 3 = \frac{27\sqrt{3} }{4} \ in^3[/tex].
For the big cake, we have
a = 4 in.
h = 4 in.
Volume of big slice = [tex]a^2 \times h \times \frac{\sqrt{3} }{4} = \frac{4^2\sqrt{3} }{4} \times 4 = 16\sqrt{3} \ in^3[/tex].
Total volume of slice = Volume of small slice + Volume of big slice
Total volume of slice = [tex]\frac{27\sqrt{3} }{4} \ in^3 +16\sqrt{3} \ in^3 = \frac{91\sqrt{3} }{4} \ in^3 = 39.404 \ in^3[/tex]
Total volume of slice = 39404 in³.
Answer:
The volume of slice is 142
The volume of the remaining cake is 708
Explanation:
