Answer:
The volume needed is 1.245 cubic millimetres.
Explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

In which
x is the number of sucesses
e = 2.71828 is the Euler number
is the mean in the given time interval.
The number of (large) inclusions in cast iron follows a Poisson distribution with a rate of 3.7 per cubic millimetre.
This means that
, in which v is the volume, in cubic millimeters.
What is the volume of material to inspect such that the probability of at least one inclusion is 0.99
Either there are no inclusions, or there is at least one. The sum of these probabilities is decimal 1. So


We want

Building the inequality


So


Applying ln to both sides



We have that:
Mean rate of 3.7 per cubic millimetre. How many cubic millimetres are 4.605?
4.605/3.7 = 1.245
The volume needed is 1.245 cubic millimetres.