Let PQRSTUVW be a rectangular prism, as shown. The area of face PQRS is 6. The area of face PQUT is 15. The area of face QUVR is 40. Find the volume of the rectangular prism.
To make the equations easier to solve, I let
QU = a
UT = b
QR = c.
Then: area(PQRS) = b·c = 6
area(PQUT) = a·b = 15
area(QUVR) = a·c = 40
Since b·c = 6 ---> c = 6/b
Since a·c = 40 ---> a = 40/c ---> a = 40/(6/b) ---> a = 40·b/6 [substituting]
Since a·b = 15 ---> (40·b/6)·b = 15 ---> 40·b2 = 90 ---> b2 = 9/4 ---> b = 3/2
Since b = 3/2 ---> b·c = 6 ---> (3/2)·c = 6 ---> c = 4
Since a·c = 40 ---> a·(4) = 40 ---> a = 10
The volume of the object = length·width·height = a·b·c = (10)·(3/2)·(4) = your answer