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 use the formulas for the volume and surface area of a rectangular block (also known as a rectangular prism or cuboid).

Step 1: Understand the problem The problem gives us the ratio of the areas of the adjacent faces of the block, which are 15:10:12. Let's denote the sides of the block as a, b, and c. Then, the areas of the adjacent faces are ab, bc, and ac. The volume of the block is given as 960 cm^3, which is equal to abc.

Step 2: Express the ratios in terms of the sides From the ratio 15:10:12, we can write the following equations: ab/15 = bc/10 = ac/12 = k (where k is a constant)

Step 3: Find the sides in terms of k From the equations in step 2, we can express the sides a, b, and c in terms of k: a = 15k/b b = 10k/c c = 12k/a

Step 4: Substitute the expressions for the sides into the volume formula The volume V of the block is given by V = abc = 960 cm^3. Substituting the expressions for a, b, and c from step 3 into this formula, we get: V = (15k/b)(10k/c)(12k/a) = 960 Solving this equation for k, we get: k^3 = 960/(151012) = 0.533333...

Step 5: Find the sides a, b, and c Substituting k = 0.533333... back into the expressions for a, b, and c from step 3, we get: a = 150.533333.../b = 8 b = 100.533333.../c = 5.33333... c = 12*0.533333.../a = 6.4

Step 6: Find the shortest edge The shortest edge of the block is the smallest of a, b, and c, which is 5.33333... cm.