The triangle $\triangle ABC$ is an isosceles triangle where $AB = 4\sqrt{2}$ and $\angle B$ is a right angle. If $I$ is the incenter of $\triangle ABC,$ then what is $BI$? Express your answer in the form $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is not divisible by any perfect square other than $1.$

Charlene's parents deposit $500 in an account on the day she is born. the account earns a high interest rate of 9.2% compounded quarterly because charlene is not allowed to access the money until her 22nd birthday. how much money will charlene have on her 22nd birthday? a: $3,698.50 b: $5,921.31 c: $101,200 d: $ 824.58