Confidence intervals are used to find a region in which we are 100 * ( 1 – α )% confident the true value of the parameter is in the interval.

For large sample confidence intervals about the mean you have:

xBar ± z * sx / sqrt(n)

where xBar is the sample mean
z is the zscore for having α% of the data in the tails, i.e., P( |Z| > z) = α
sx is the sample standard deviation
n is the sample size

To find the sample size needed for a confidence interval of a given size we need only to concern ourselves with the error term of the CI.
We know that the interval is centered at xbar so we need to find the value of n such that

z * sx / sqrt(n) = width.

The z-score for a 0.95 confidence interval is the value of z such that 0.025 is in each tail of the distribution.
z= 1.959964

The equation we need to solve is: z * sx / sqrt(n) = width

n = (z * sx / width) ^ 2.

n = ( 1.959964 * 400 / 50 ) ^ 2
n = 245.8534

Since n must be integer valued we need to take the ceiling of this solution. Always take the ceiling so that the size of the CI will be correct.

n = 246