The tree faces of a rectangular solid have areas 6,10, and 15. If the dimensions of the rectangular solid are all integers, what is the volume of the solid?
The dimensions must be 2, 3, and 5.
Multiply those three numbers together to find the volume.
To find the volume of the rectangular solid, we need to determine its dimensions. Let's call the length, width, and height of the rectangular solid x, y, and z, respectively.
We know that the areas of the faces are 6, 10, and 15.
The area of the first face is xy = 6.
The area of the second face is xz = 10.
The area of the third face is yz = 15.
Now, let's look for integer values of x, y, and z that satisfy these equations.
From the first equation, xy = 6, we can list the possible pairs (x, y) of factors of 6:
(1, 6), (2, 3), (3, 2), (6, 1)
From the second equation, xz = 10, we can list the possible pairs (x, z) of factors of 10:
(1, 10), (2, 5), (5, 2), (10, 1)
From the third equation, yz = 15, we can list the possible pairs (y, z) of factors of 15:
(1, 15), (3, 5), (5, 3), (15, 1)
Now, we need to find a combination of these pairs that works.
If we examine the pairs, we can see that the only combination that works is:
(x, y, z) = (2, 3, 5)
Therefore, the volume of the rectangular solid is x * y * z = 2 * 3 * 5 = 30.
The volume of the solid is 30 cubic units.