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I understood how I got the $1.50 however could not understand ‘the next 38,000 direct labour hours’
The shadow price is $1.50 because each extra hour of labour they get will mean they can produce an extra 0.25 units of Y.
However once the have enough labour to produce 12,000 units of Y (together with the 10,000 units of X) they will not want to buy any more labour because the demand for Y is only 12,000 units. At that level of production they will be using 98,000 hours which is 38,000 hours more than is currently available.
still cannot understand how did you get to 98,000 hours (that is the 38,000 hours extra)
You know from the labour constraint given in the question that each unit of X takes 5 hours, and each unit of Y takes 4 hours.
If they produce 10,000 units of X and 12,000 units of Y (for the reasons I have already explained). then the total labour hours is 98,000.
Have you watched my lectures on linear programming?
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