Problem 4 19 are often used as a component in engineered impervious barriers for

Question

Problem 4 19 are often used as a component in engineered impervious barriers for haz Geomembranes ardous waste containment 20 For example, geomembranes are used to line the bottom of pits in which waste is to be buried 21 Geomembrane materials are delivered to a site as large rolls (see Figure 5). The geomem- Figure 5: Geomembrane liners are thin plastic or rubber sheets. They are manufactured and delivered as large rolls. [http://livablecranbrook.blogspot.com] brane rolls may be long, but they are NOT wide enough to cover the entire bottom of a pit or pond (see Figure 6). As such, the long sheets must be "sewn" together to cover the site The resulting seams are the weakest link in the engineered impervious barriers 22

Explanation / Answer

Solution

Back-up Theory

If X = number of defects per length L of a product, Y = number of defects per length kL of the same product, and X ~ Poisson with parameter ,

then Y ~ Poisson with parameter k………………………………………………………(1)

P(X = 0) = e- and P(Y = 0) = e-k ……………………………………………………….(2)

Now, to work out the solution,

Let X = Number of undetected defects

Part (a)

Given total seam length = 5000 m, Number of seam defects per 250 m = 1, and 10% of seam defects go undetected,

Mean rate of undetected defects = (1 x 0.1) = 0.1 per 250 m or 1 per 2500 m ANSWER

Part (b)

Probability of 4 or more undetected defects in the project length = P(X 4) under = 2 per 5000 m [vide (1) under Back-up Theory]

= 1 – P(X < 4) = 1 – 0.8571 = 0.1429 ANSWER [probability is obtained using Excel Function]

Part (c)

Here, we have seam length as 1000 m and we want P(X = 0) 0.99.

Now, applying (1) under Back-up Theory, we should have e- 0.99 or taking natural logarithm (ln) of both sides, - ln(0.99) = - 0.01005 or

< 0.01005 [note that division by – 1 would reverse the inequality sign]

So, the maximum allowable fraction of undetected defects can only be 1% ANSWER

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