MHB Calculate the length of the fourth side edge.

[ghostfirefox]

The three lateral edges of a pyramid based on a square are 6016, 2370 and 4350 long. Calculate the length of the fourth lateral edge. We assume that edges 6016 and 2370 extend from the opposite tops of the base.

 

[skeeter]

The three lateral edges of a pyramid based on a square are 6016, 2370 and 4350 long. Calculate the length of the fourth lateral edge. We assume that edges 6016 and 2370 extend from the opposite tops of the base.

let the four corners of the square base with side length $s$ lie in the x y plane with positions

$(0,0,0)$, $(s,0,0)$, $(s,s,0)$, and $(0,s,0)$

let the apex of the pyramid be at position $(a,b,c)$

... assume that edges 6016 and 2370 extend from the opposite tops of the base

using the distance formula between two points in space yields the following equations

$a^2+b^2+c^2 = 6016^2$
$(a-s)^2+(b-s)^2+c^2 = 2370^2$
$(a-s)^2+b^2+c^2=4350^2$
$a^2+(b-s)^2+c^2 = d^2$, where $d$ is the length of the fourth edge

use the system of equations to solve for $d$ ... I get a unique integral value for $d$ such that $4350 < d < 6016$

 

[ghostfirefox]

I tried to calculate this, but there are too many variables in this system of equations.

 

[skeeter]

Subtract the 3rd equation from the 1st.

Subtract the 2nd equation from the 4th.

Work the solution for $d$ from the resulting two equations.