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Although as originally designed, the Mark 33 did not have radar, many were retrofitted with sets during the war. For example, see the second picture at this link:

NavWeaps 5"/38 pictures page

I paraphrase some information from "Naval Weapons of World War Two" by Campbell for the following:

This was the first powered director (the previous Mark 19 and Mark 28 were hand-driven). Designed for target speeds up to 275 knots, later increased to 320 with some capability against diving targets at up to 400 knots. Stable element, stereo RF and analog computer were all included in the single-pedestal installation. "Early directors were open-topped but shields and radar were later fitted with some difficulty in balancing."

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Quoting from "US Naval Weapons" by Friedman:

Mk 33 represented the limit of complexity which could be mounted in a single light, fast-turning director. As long as computing remained mechanical, greater target speed had to require larger systems which in turn increased weight at the director and made rapid movements more difficult. In 1936, the Bureau of Ordnance began work on a solution, Mk 37, in which the computer was moved below decks, leaving a more compact (and better protected) position above, dedicated solely to tracking targets and to rangefinding.

A couple of other comments, based upon my knowledge of these systems:

The Mark 33 was first mounted afloat in 1934 aboard the carrier USS Ranger and mods were available to control both 5"/25 and 5"/38 mountings (basically different "cams" in the computer to account for the different ballistics of the two weapons).

These systems operated in a 3-D environment, unlike the 2-D environment of the British HACS AA control system. They were "tachymetric," to use the British term for 3-D control. One output of the computer was back to the director, which allowed the director trainer and spotter to check on the accuracy of the computation - in other words, if this output kept the director on target, then the solution was "good". Another output went to the guns, effectively the "future position" information that included the gun train, elevation and fuze setting data. All US systems used stereo RFs, as these did not need a prominent feature on the target in order to get a good range finding - Co-I RF systems require a easily-distinguishable feature, such as a ship's mast, which can be split and brought together in order to get an accurate range reading, something difficult to do on a rapidly moving aircraft target. Stable verticals (gyro compensation for the ship's yaw, pitch and roll) were incorporated into the Mark 33 as were optical stabilization for the RF.

Successive "cuts" from the RF were fed into the computer, from which it could derive the target's current course and speed (in 3-D) and thus predict its future position.