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(A) Let the total age of $10$ directors be $T_1 = 10 	imes 48 = 480$ years.Let the age of the director who died be $x$ years.When one director aged $

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$56$

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(A) Let the total age of $10$ directors be $T_1 = 10 	imes 48 = 480$ years.Let the age of the director who died be $x$ years.When one director aged $53$ resigned and one director aged $x$ died,and a new director aged $34$ joined,the number of directors became $9$.The total age of these $9$ directors at that moment was $T_2 = 480 - 53 - x + 34 = 461 - x$.One year later,each of the $9$ directors aged by $1$ year,so the total age increased by $9 	imes 1 = 9$ years.The new total age is $T_3 = (461 - x) + 9 = 470 - x$.We are given that the average age of these $9$ directors after one year is $46$ years.Therefore,$T_3 = 9 	imes 46 = 414$.Equating the two expressions for $T_3$: $470 - x = 414$.Solving for $x$: $x = 470 - 414 = 56$.Thus,the age of the deceased director was $56$ years.

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