The air conditioning unit has a cooling coil which discharges the air at condition CC (CC in this case is 52db,51wb). Then, there is a reheat coil which performs simple heating (constant humidity ratio) until the desired room supply condition is achieved. Let's call this condition (S). We only know S will be to the right of CC, so locate CC on the psych chart and draw a horizontal line through CC. This horizontal line is the reheat process, and is shown in green in the chart below. S will be somewhere on this line to the right of CC.
You are also given the room (or return) air condition which we will call (R) and is 75db, 50%rh. We know the air process as it goes from the supply registers to the return grills is a straight line from (S) to (R) with a slope given by the SHR of 0.66. Therefore, locate (R) on the psych chart and draw through it a line with SHR=0.66, which is shown as a dark blue line in the chart (you need a psych chart with a protractor). We know (S) will be on this line to the left of (R).
Therefore, (S) will be at the intersection of these two lines. That's how you obtain the 62F. You can calculate the heat required for the CC-S process by reading the enthalpy change directly from the chart, or by using an equation like they did in their solution.
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