Solution:

Given that 50% of the product has weight 60 Kg or more, and 10% have weight 55 Kg or less. Let us assume that the weight of the product follows a normal distribution.

Finding the Mean:

We know that the normal distribution is symmetric, and the mean is the midpoint of the distribution. Therefore, the mean of the distribution is:

Mean = (60 + 55) / 2 = 57.5 Kg

Finding the Standard Deviation:

Using the empirical rule, we can say that:

- 68% of the product falls within one standard deviation of the mean
- 95% of the product falls within two standard deviations of the mean
- 99.7% of the product falls within three standard deviations of the mean

Let us assume that x is the weight of the product and σ is the standard deviation.

- 50% of the product have weight 60 Kg or more. This implies that:

P(x ≥ 60) = 0.50

Using the standard normal distribution table, we can find the value of the z-score:

z = (60 - 57.5) / σ = 2.5 / σ

From the table, we get:

P(z ≥ 0.67) = 0.2514

Therefore, P(x ≥ 60) = 0.2514

- 10% of the product have weight 55 Kg or less. This implies that:

P(x ≤ 55) = 0.10

Using the standard normal distribution table, we can find the value of the z-score:

z = (55 - 57.5) / σ = -2.5 / σ

From the table, we get:

P(z ≤ -0.83) = 0.2033

Therefore, P(x ≤ 55) = 0.2033

Adding both probabilities, we get:

P(x ≤ 55) + P(x ≥ 60) = 0.2033 + 0.2514 = 0.4547

Using the complement rule, we get:

P(55 < x="" />< 60)="1" -="" p(x="" ≤="" 55)="" -="" p(x="" ≥="" 60)="1" -="" 0.4547="" />

Using the standard normal distribution table, we can find the value of the z-score:

z = (57.5 - 57.5) / σ = 0

From the table, we get:

P(-z < z="" />< z)="P(-1" />< z="" />< 1)="" />

Therefore, P(55 < x="" />< 60)="0.5453" ≈="" />

Solving for σ, we get:

(60 - 57.5) / σ = 0.84

σ ≈ 2.98 Kg

Finding the Variance:

Variance = σ^2 = (2.98)^2 = 8.88 Kg^2

Given (1.28) = 0.90:

The value (1.28) is the z-score corresponding to the cumulative probability of 0.90. From the standard normal distribution table, we get:

P(z ≤ 1.28) = 0.8997

Therefore, we can say that 90% of the product have