It is known that if Z has a normal distribution with mean μ and standard deviation σ, then (Z - μ) / σ has the standard normal distribution, i.e., the one with mean 0 and standard deviation 1. We need to find Δ such that P(μ - Δ ≤ Z ≤ μ + Δ) = .9. Verify that μ - Δ ≤ Z ≤ μ + Δ is equivalent to -Δ / σ ≤ (Z - μ) / σ ≤ Δ / σ.

Since the standard normal distribution is symmetric, P(-Δ / σ ≤ (Z - μ) / σ ≤ Δ / σ) = 2P(0 ≤ (Z - μ) / σ ≤ Δ / σ). Now use the table such as this one to find Δ / σ such that 2P(0 ≤ (Z - μ) / σ ≤ Δ / σ) = .9. Since σ is given, you can find Δ.