The areas of the adjacent faces of a rectangular block are in the ratio 15 : 10 : 12 and its volume is 960 cm3. Find the length of its shortest edge.

To solve this problem, we need to follow these steps:

1. Let's denote the sides of the rectangular block as a, b, and c. The areas of the adjacent faces are then ab, bc, and ac. According to the problem, these areas are in the ratio 15:10:12.

2. We can write this ratio as ab/15 = bc/10 = ac/12 = k, where k is a constant.

3. From these equations, we can express b = 15k/a and c = 10k/b. Substituting b into the second equation gives c = 10k/(15k/a) = 2a/3.

4. The volume of the rectangular block is given by V = abc = 960 cm^3. Substituting b and c into this equation gives a*(15k/a)*(2a/3) = 960.

5. Simplifying this equation gives 10ka = 960, so k = 96/a.

6. Substituting k back into the equations for b and c gives b = 15*96/a = 1440/a and c = 2a/3.

7. The shortest edge of the rectangular block is the smallest of a, b, and c. Since a and b are inversely proportional and c is directly proportional to a, the smallest edge will be c when a is at its maximum value.

8. The maximum value of a occurs when a = b, so 1440/a = a. Solving this equation gives a = sqrt(1440) = 38 cm (approximately).

9. Substituting a = 38 cm into the equation for c gives c = 2*38/3 = 25.3 cm (approximately).

So, the length of the shortest edge of the rectangular block is approximately 25.3 cm.