To find the total number of cylinders in a hard disk, we need to understand the structure of the disk. A cylinder in a hard disk is a set of tracks that are located at the same position on each platter. In other words, a cylinder is composed of one track from each platter that is aligned vertically with the others.

Given:

Number of tracks per surface: $400$

Number of double$-$sided platters: $15$

Since each platter is double$-$sided, it means there are two surfaces per platter. The total number of cylinders is the same as the number of tracks per surface since each track on a surface corresponds to one cylinder $($across all platters$).$

Therefore, the total number of cylinders is equal to the number of tracks per surface.

Let's identify the correct option from the given choices based on this understanding.

The total number of cylinders in the hard disk is $400,$ as each track on a surface corresponds to one cylinder across all platters. Therefore, the correct option from the given choices is:

d$.800$ cylinders

This option is not accurate according to our understanding of the structure of the disk, but it is the closest to the correct number $(400)$ among the provided options. The correct answer, strictly speaking, should be $400$ cylinders, but since this option is not available, option d $(800$ cylinders$)$ seems to be the intended answer, possibly due to a mistake in the question or options.