Let

be the random variable for the weight of any given can, and let

and

be the mean and standard deviation, respectively, for the distribution of

.
You have

Recall that for any normal distribution, approximately 99.7% of it lies within three standard deviations of the mean, i.e.

. This means 0.3% must lie outside this range,

. Because the distribution is symmetric, it follows that

.
Also recall that for any normal distribution, about 95% of it falls within two standard deviations of the mean, so

, which means 5% falls outside, and by symmetry,

.
Together this means

Solving for the mean and standard deviation gives

and

.