[THOUGHT] Let me solve this step by step.

For a confidence interval for the population mean when the population standard deviation is known:

• Population standard deviation σ = 5
• Sample size n = 36
• Confidence level = 99%
• z-value for 99% confidence = 2.58 (given that P(|Z| ≤ 2.58) = 0.99)

The confidence interval formula is:

The margin of error E = z × (σ/√n) = 2.58 × (5/√36) = 2.58 × (5/6)

Given that the lower bound is 1.2, and the upper bound is a:

• Lower bound = x̄ - E = 1.2
• Upper bound = x̄ + E = a

The confidence interval is symmetric around the sample mean, so:

• x̄ = (1.2 + a) / 2
• E = (a - 1.2) / 2

Also, E = 2.58 × (5/6)

So: (a - 1.2) / 2 = 2.58 × (5/6)

Let me calculate this. [/THOUGHT] [PYTHON] import math

Given values

sigma = 5 # population standard deviation n = 36 # sample size z = 2.58 # z-value for 99% confidence

Calculate margin of error

E = z * (sigma / math.sqrt(n)) print(f"Margin of error E = {E}")

Given lower bound is 1.2

lower_bound = 1.2

The sample mean is at the center of the interval

lower_bound = x_bar - E

So x_bar = lower_bound + E

x_bar = lower_bound + E print(f"Sample mean x_bar = {x_bar}")

Upper bound a = x_bar + E

a = x_bar + E print(f"Upper bound a = {a}")

Calculate 10a

result = 10 * a print(f"10a = {result}") [/PYTHON]

[THOUGHT] The calculation is complete. Let me verify:

• Margin of error E = 2.58 × (5/6) = 2.15
• Lower bound = 1.2
• Sample mean x̄ = 1.2 + 2.15 = 3.35
• Upper bound a = 3.35 + 2.15 = 5.5

Therefore, a = 5.5 and 10a = 55. [/THOUGHT]

$\boxed{55}$