Let's determine the correct function that models the exponential change between the number of months, [tex]\( t \)[/tex], and the amount of money deposited in the savings account, [tex]\( a(t) \)[/tex].
We know the following:
- The initial deposit is [tex]\( \$50 \)[/tex].
- The amount of money in the account triples each month.
Given this information, we need an exponential function that represents the amount of money after [tex]\( t \)[/tex] months. In general, an exponential growth function can be written as:
[tex]\[ a(t) = P \cdot r^t \][/tex]
where:
- [tex]\( P \)[/tex] is the initial amount (in this case, \[tex]$50),
- \( r \) is the growth rate (in this case, the amount triples, so \( r = 3 \)).
Now, substituting these values into the general form of the exponential function:
\[ a(t) = 50 \cdot 3^t \]
So, the correct function that models the growth of the savings account is:
\[ a(t) = 50 \cdot 3^t \]
Thus, the correct choice is:
\[ a(t) = 50 \cdot 3^t \]
If we want to know the amount in the account after 2 months, we substitute \( t = 2 \) into the function:
\[ a(2) = 50 \cdot 3^2 = 50 \cdot 9 = 450 \]
After 2 months, the amount in the savings account will be \$[/tex]450.
