Question 42303
At 9 am the price is $65.00.


Each hour the price decreases by 10% of its value in the previous hour.


After {{{red(1)}}} hr the price will be $(65.00 - 0.1x65.00) = $({{{0.9^(red(1))}}}x65.00).


So after 2 hours the price will be 10% less than the price after 1 hour.
After {{{red(2)}}} hrs the price will be $(0.9x65.00 - 0.1x0.9x65.00) = $({{{0.9^(red(2))}}}x65.00).


Thus, after {{{red(3)}}} hrs the price will be $({{{0.9^2}}}x65.00 - 0.1x{{{0.9x0^2}}}.1x65.00) = $({{{0.9^(red(3))}}}x65.00).


Similarly, after {{{red(n)}}} hrs the price will be $({{{0.9^(red(n))}}}x65.00).


[Note the similarity between the numbers marked in red ink]


Let us assume that the price becomes $46 or less after 'n' hours from 9 am.


Then, {{{0.9^n}}}{{{x65 < 46}}}
or {{{0.9^n < 46/65}}}
or {{{0.9^n < 0.708}}} ______(1)


By trial, 
when n = 2, LHS = 0.81 > RHS
when n = 3, LHS = 0.729 > RHS
when n = 4, LHS = 0.6561 < RHS


These value of 'n' satisfies the inequation (1) and so Amy will be able to purchase the dress at price below $46 after 4 hrs from 9 am.
Then the price of the dress will be $({{{0.9^4}}}x65.00) = $42.65 (approx).