To find out how much the man earned during the week when he worked 52 hours, we can break down his earnings into two parts: for the first 40 hours and for the overtime hours.
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Earnings for the first 40 hours: The man earns $1.80 per hour for the first 40 hours. \[ \text{Earnings for 40 hours} = 40 \text{ hours} \times 1.80 \text{ dollars/hour} = 72 \text{ dollars} \]
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Calculation of overtime rate: The man earns \( \frac{1}{12} \) of his regular hourly rate for the hours worked over 40. His regular hourly rate is $1.80, so his overtime rate will be: \[ \text{Overtime Rate} = \frac{1}{12} \times 1.80 = 0.15 \text{ dollars/hour} \]
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Earnings for the overtime hours: He worked 52 hours in total, which means he worked: \[ \text{Overtime hours} = 52 \text{ hours} - 40 \text{ hours} = 12 \text{ hours} \] Therefore, his earnings for the 12 hours of overtime will be: \[ \text{Earnings for overtime} = 12 \text{ hours} \times 0.15 \text{ dollars/hour} = 1.80 \text{ dollars} \]
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Total earnings for the week: Finally, we add the earnings from the regular hours and the overtime hours: \[ \text{Total Earnings} = \text{Earnings for 40 hours} + \text{Earnings for overtime} = 72 \text{ dollars} + 1.80 \text{ dollars} = 73.80 \text{ dollars} \]
Thus, the total amount of money he earned during the week when he worked 52 hours is:
\[ \boxed{73.80} \]