Question

Civil engineers believe that W, the amount of weight (in units of 1000 pounds) that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with mean 400 and standard deviation 40. Suppose that the weight (again, in units of 1000 pounds) of a car is a random variable with mean 3 and standard deviation .3. Approximately how many cars would have to be on the bridge span for the probability of structural damage to exceed 0.1?

Answer 1

Answer:

x  ≥ 117

Step-by-step explanation:

Let suppose Pₙ denoted the structural damage that is probable with n cans

Pₙ = P {X₁+ ....... + Xₙ ≥ w}

    = P {X₁+ ....... + Xₙ-w ≥ 0 }

So weight of the ith can denoted by Xi

From CLT , ΣXi is normal with 3n and variance 0.09

w is independent of Xi and normal

So

ΣXi-w is also normal

then mean and variance are

E [ ΣXi - w] = 3n -400

Var (ΣXi-w) = var (ΣXi) +Var (w)

                  = 0.09n +1600

So Pₙ = P{ X₁ +X₂ +.... +Xₙ -w-(3n-400) /underroot 0.09n +1600 ≥ -(3n-400)/ understoot (0.09n +160v)

this can be simplified to≈ P₀ { Z ≥ 400 - 3n / undersoot (0.09n+1600)}

P{Z ≥ 1.28 } ≈ .1

on simplifying x  ≥ 117