Short Answer 
The value of x is determined to be 7 through the application of triangle inequality rules and the cosine rule, ensuring the triangle remains acute. The inequalities established that x must be greater than 6.7 and less than 7.5, with 7 being the only valid solution.
Step 1: Establish Triangle Conditions
To find the value of x in the triangle with sides x cm, 2x cm, and 15 cm, we first need to apply the triangle inequality rules. These rules state:
- The sum of the two smallest sides must be greater than the third side.
- The difference between the two smallest sides must be less than the third side.
From the inequalities, we derive:
- Sum: x + 2x > 15 leads to x > 5.
- Difference: 2x – x < 15 leads to x < 15.
Step 2: Apply Cosine Rule for Acute Triangle
Next, we use the cosine rule to ensure the triangle is acute. We have:
- Let a = x, b = 2x, c = 15.
- The formula is cos A = (b² + c² – a²) / (2bc).
- For the triangle to be acute, cos A > 0, leading to the inequality 5x² – 225 > 0.
This simplifies to x > 6.7 (approximate) to maintain the acute angle condition.
Step 3: Final Determination of x’s Value
Combining the conditions from both steps provides a range for x. We have:
- From Step 1: x > 5 and x < 15.
- From Step 2: x > 6.7 and x < 7.5.
With x needing to satisfy all conditions, the only value that fits is x = 7, confirming our conclusion.