A rectangular prism, of which the base is a square, has a diagonal of length d, and the total area of all its faces is b. In terms of b and d, what is the total length of all its edges?
Let x = side of square base,
and y = height of prism
And assume that the prism is a right rectangular prism, or the the base and the height are perpendicular.
In the square base:
diagonal = d .....given
So, by Pythagorean theorem, d^2 = x^ +x^2
x^2 = (d^2) /2
x = d / (sqrt(2))
Area of all the faces of the prism:
A = 2(x^2) +4(x*y) = b .....given
2x^2 +4xy = b
4xy = b -2x^2
y = (b -2x^2) / (4x)
y = [b -2(d^2 / 2)] / [4(d / sqrt(2)]
y = [b -d^2] / [2sqrt(2) *d]
Simplify further if you like...
Therefore, the sum of all the edges of this prism is
= 8x +4y
= [8(d / sqrt(2)] +[4(b -d^2) / (2sqrt(2) *d)]
= [4sqrt(2) *d] + [2sqrt(2) *(b -d^2)/d]
= 2sqrt(2)*[2d +(b -d^2)/d]
= 2sqrt(2)[(2d^2 +b -d^2) / d]
= 2sqrt(2)[(b +d^2)/d] -----------------answer.