3. A random force P is applied to tow a crate of W ~ N(50, 1.5²) kg. P is normally distributed with a standard deviation σ_P=5. P and W are independent. What is the minimum mean of P so that the probability of towing the crate is greater than 99.99%?

Solution

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Solving the system of equations, with W ~ N(50, 1.5²) kg, we can obtain the distributions of P and N

Thus, we have obtained the distributions of the normal force and the external force P: N ~ N(368, 11²) N and S ~ N(153.5, 4.6²) N. Ans.

Thus, we can construct

And

The probability of towing the crate can be calculated as

Solve the above equation inversely, we can obtain the minimum mean μ_P=179 N. Ans.