Jim has a triangular shelf system that attaches to his showerhead. The total height of the system is 18 inches, and there are three parallel shelves as shown above. What is the maximum height, in inches, of a shampoo bottle that can stand upright on the middle shelf?

Answer

9

Explanation

Since the three shelves of the triangular shelf system are parallel, the three triangles in the figure are similar. Since the shelves divide the left side of the largest triangle in the ratio 2 to 3 to 1, the similarity ratios of the triangles are as follows
Smallest to middle : 2 to 5
Smallest to largest : 2 to 6, or 1 to 3
Middle to largest : 5 to 6
The height of the largest shampoo bottle that can stand upright on the middle shelf is equal to the height of the middle shelf. The height of the entire triangular shelf system is 18 inches. This is the height of the largest triangle. The height of the middle shelf is the height of the middle triangle minus the height of the smallest triangle. Since the similarity ratio of the middle triangle to the largest triangle is 5 to 6, the height of the middle shelf is `(5)/(6)` (18) = 15 inches. Since the similarity ratio of the smallest triangle to the largest triangle is 1 to 3, the height of the middle shelf is `(1)/(3)` (18) = 6 inches. Therefore, the height of the middle shelf is 9 inches.