Question

example 4 exponential decay
banks in banking, a dormant account is one that has not been used
in over a year. a bank charges a monthly fee on dormant accounts of
0.8% of the account balance. one dormant account initially had a
balance of $1609.
part a write an equation to represent the balance in the account after
t months.
$y = a(1 - r)^t$ \t\t\tequation for exponential decay
$y = \underline{\quad\quad}(1 - \underline{\quad\quad})^t$ \t$a = 1609$ and $r = 0.8\\%$ or $0.008$
$y = \underline{\quad\quad}(\underline{\quad\quad})^t$ \t\tsimplify.
the equation is \underline{\quad\quad\quad\quad\quad\quad}, where $y$ is the balance in
the account after $t$ months.
part b estimate the balance in the account after a year.
substitute \underline{\quad\quad} for $t$ in the equation.
using $y = 1609(0.992)^t$, the balance in the account after a year
is about \underline{\quad\quad}.

Explanation:

Step1: Substitute initial value & rate

$y = 1609(1 - 0.008)^t$

Step2: Simplify the decay factor

$y = 1609(0.992)^t$

Step3: Substitute t=12 for 1 year

$y = 1609(0.992)^{12}$

Step4: Calculate the final balance

First compute $0.992^{12} \approx 0.9083$, then $1609 \times 0.9083 \approx 1461.45$

Answer:

Part A:

The equation is $\boldsymbol{y = 1609(0.992)^t}$, where $y$ is the balance in the account after $t$ months.

Part B:

The balance in the account after a year is about $\boldsymbol{\$1461.45}$