Final answer:
We used the proportion a:b:c = 5:3:2 to express a, b, and c in terms of a constant k, substituted these expressions into the given equation to solve for k, and then found the values of a, b, and c as 15, 9, and 6 respectively.
Step-by-step explanation:
The student is given the proportional relationship a:b:c = 5:3:2 and the equation a²-b²-c²=108. To find the values of a, b, and c, we must first express a, b, and c in terms of a single variable using the given ratios. Let's say a = 5k, b = 3k, and c = 2k, where k is a positive constant. Substituting these into the equation, we get (5k)² - (3k)² - (2k)² = 108, which simplifies to 25k² - 9k² - 4k² = 108, and further to 12k² = 108.
Solving for k, we find,
k² = 9, and since k > 0, k = 3.
We can then find the values of a, b, and c:
a = 5k = 15, b = 3k = 9, and c = 2k = 6.