A bank account is opened with an initial deposit of 800. The account balance increases by … — SAT

A bank account is opened with an initial deposit of $800. The account balance increases by$ 5\% $each year. If no other transactions are made, which of the following functions best models the account balance,$ B(t) $, in dollars, after$ t$ years?

$B(t) = 800(1.05)^t$ Answer
B
$B(t) = 800(0.05)^t$
C
$B(t) = 800(1.5)^t$
D
$B(t) = 800 + 40t$

Answer

The function

B(t) = 800(1.05)^t

best models the account balance.

The initial deposit of

800 represents the initial value of the exponential function when

t = 0

. Since the account balance increases by

5\%

each year, the value is multiplied by a growth factor of

1 + 0.05 = 1.05

each year. Thus, the exponential function that models the balance after

years is

B(t) = 800(1.05)^t$.

Step-by-Step Solution

Identify the initial value of the exponential growth function.

The initial deposit is

800, so the initial value at

t = 0

is

800$.

An exponential model is written in the form

B(t) = a(b)^t

, where

a

represents the initial value.

Determine the growth factor based on the annual percentage increase.

The growth rate is

r = 5\% = 0.05

. The growth factor

b

1 + r = 1 + 0.05 = 1.05

To find the growth factor for an increasing quantity, add the growth rate as a decimal to 1.

Write the final exponential function by substituting the initial value and growth factor.

The function is

B(t) = 800(1.05)^t

Substitute

a = 800

and

b = 1.05

into the standard exponential form

B(t) = a(b)^t

Key Concept

Writing and interpreting exponential growth functions from a real-world context.

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