Writing and Solving Exponential Equations: Mastery Test
Enter the correct answer in the box.
Susanna deposits [tex]$\$ 400$[/tex] in a savings account with an interest rate of [tex]3 \%[/tex] compounded annually. What equation could Susanna use to calculate how many years it will take for the value of the account to reach [tex]$\[tex]$ 600$[/tex][/tex]?
Fill in the values of [tex]A[/tex], [tex]D[/tex], and [tex]c[/tex] to write the equation modeling this situation.
To solve the problem of determining how many years it will take for Susanna's deposit to grow from \[tex]$400 to \$[/tex]600 with an annual interest rate of 3%, we can follow these steps:
1. Identify the formula for compound interest:
[tex]\[ A = P(1 + r)^t \][/tex]
Where: - [tex]\( A \)[/tex] is the target amount (in this case, \[tex]$600),
- \( P \) is the principal (initial deposit, \$[/tex]400), - [tex]\( r \)[/tex] is the annual interest rate (3% or 0.03), - [tex]\( t \)[/tex] is the time in years.
2. Set up the equation with the given values:
[tex]\[ 600 = 400(1 + 0.03)^t \][/tex]
3. Simplify the equation:
[tex]\[ 600 = 400(1.03)^t \][/tex]
4. Isolate the exponential term:
[tex]\[ \frac{600}{400} = (1.03)^t \][/tex]
[tex]\[ 1.5 = (1.03)^t \][/tex]
5. Take the natural logarithm (ln) of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln(1.5) = \ln((1.03)^t) \][/tex]
6. Use the property of logarithms to bring the exponent [tex]\( t \)[/tex] in front:
