Maths Help?

I interpret "slant edge" to mean one of the edges with an endpoint at E. I wouldn't call any of the base edges a "slant edge", but I'm in the US, you're in the UK, and I have no idea if that could be important.

So a "slant edge" would be one of the edges EA, EB, EC, or ED. They're identical in length because it's a right pyramid.

The line EV would NOT be a "slant edge", because EV is not an edge.


If we assume my interpretation of "slant edge" is correct, it's still unclear to me what 3e is asking for. It could be the angle EBU (or ECU, etc.), or it could be the angle EBA (or EBC, etc.). Either one could be seen as fitting the description of 3e. I would probably calculate both, label them clearly in my answer, and then write a brief note that the question could use some clarification.

 

EV is the distance between the vertex and the middle of one of the sides at the base, the slant edge connects the vertex E to one of the corners of the base (like EB).
same with the angles (EVU vs EBU).

 

a) Volume = 1/3 x Area of base x height
= 1/3 x 42mm x 42mm x 68mm
= 39 984 mm^3

b) Pythagoras Theorem
c^2 = a^2 + b^2
where c = EV, a = EU b= UV
c = √(68^2 + 21^2)
c = √ (5065mm^2)
c = 71.16881mm
EV = 71.16881mm


c) Angle EVU = arc tan or tan^-1 (opposite/adjacent)
Opposite = line EU Adjacent = line UV
Angle EVU = tan^-1 (68mm/21mm)
Angle EVU = 72.838 degrees or 1.27126434 radians

d) Pythagoras Theorem to find length UB
c2 = a2 + b2
where c = UB a = UV b = UW where W = midpoint of line AB
c = √(21^2 + 21^2)
c = 29.69848mm
UB = 29.69848mm

Pythagoras Theorem to find length EB
where c = line EB a = line EU b = line UB
c^2 = a^2 + b^2
c = √(68^2 + 29.69848^2)
c = 74.20243mm
EB = 74.20243mm

e) Angle EBU
Find BU length using Pythagoras theorem
where c = UB a = UV b = UW where W = midpoint of line AB
c^2 = a^2 + b^2
c = √(21^2 + 21^2)
c = √ (882mm^2)
c = 29.69848mm
line UB = 29.69848mm

Angle EBU = arc tan or tan-1 (opposite/adjacent)
where opposite = EU @ 68mm, adjacent = UB @ 29.69848mm

= tan-1 (68mm/ √ (882mm^2) )
Angle EBU = 66.4070684 degrees or 1.15902199 radians

f) Side length of cube = Cube root or 3√ volume of pyramid from answer a)
= 3√ 39 984 mm^3
= 34.19496 mm


Round off or convert the units to cm in the answers if necessary .