Question
Problem Description:
"I received $2,500 for my birthday and opened a savings account. The amount is expected to increase 2% each year. Write a function that models the situation where t represents the number of years. How much money will I have in 3 years? (3 points)"
Questions/Fill-in-the-Blanks:
$f(x) = \boxed{\text{ }}^\boxed{\text{ }}$
In 3 years the account will have
"I received $2,500 for my birthday and opened a savings account. The amount is expected to increase 2% each year. Write a function that models the situation where t represents the number of years. How much money will I have in 3 years? (3 points)"
Questions/Fill-in-the-Blanks:
$f(x) = \boxed{\text{ }}^\boxed{\text{ }}$
In 3 years the account will have
Answers
GPT-4o mini
To model the situation where the amount of money in the savings account increases by 2% each year, we can use the formula for exponential growth:
\[ A(t) = A_0 \times (1 + r)^t \]
where:
- \( A(t) \) is the amount after \( t \) years,
- \( A_0 \) is the initial amount (in this case, $2,500),
- \( r \) is the growth rate (2% or 0.02), and
- \( t \) is the number of years.
Plugging in the values, we have:
1. Initial amount \( A_0 = 2500 \)
2. Growth rate \( r = 0.02 \)
The function can be represented as:
\[ A(t) = 2500 \times (1 + 0.02)^t \]
\[ A(t) = 2500 \times (1.02)^t \]
Now, we can fill in the blanks:
- For the function in the format you provided:
\[ f(t) = 2500 \times (1.02)^t \]
To find out how much money will be in the account after 3 years, we substitute \( t = 3 \):
\[ A(3) = 2500 \times (1.02)^3 \]
Calculating \( (1.02)^3 \):
\[ (1.02)^3 = 1.061208 \]
Thus:
\[ A(3) = 2500 \times 1.061208 \]
\[ A(3) \approx 2653.02 \]
So in 3 years, the account will have approximately $2,653.02.
The final answers are:
- \( f(t) = \boxed{2500} \times \boxed{(1.02)^t} \)
- In 3 years the account will have \(\boxed{2653.02}\)
\[ A(t) = A_0 \times (1 + r)^t \]
where:
- \( A(t) \) is the amount after \( t \) years,
- \( A_0 \) is the initial amount (in this case, $2,500),
- \( r \) is the growth rate (2% or 0.02), and
- \( t \) is the number of years.
Plugging in the values, we have:
1. Initial amount \( A_0 = 2500 \)
2. Growth rate \( r = 0.02 \)
The function can be represented as:
\[ A(t) = 2500 \times (1 + 0.02)^t \]
\[ A(t) = 2500 \times (1.02)^t \]
Now, we can fill in the blanks:
- For the function in the format you provided:
\[ f(t) = 2500 \times (1.02)^t \]
To find out how much money will be in the account after 3 years, we substitute \( t = 3 \):
\[ A(3) = 2500 \times (1.02)^3 \]
Calculating \( (1.02)^3 \):
\[ (1.02)^3 = 1.061208 \]
Thus:
\[ A(3) = 2500 \times 1.061208 \]
\[ A(3) \approx 2653.02 \]
So in 3 years, the account will have approximately $2,653.02.
The final answers are:
- \( f(t) = \boxed{2500} \times \boxed{(1.02)^t} \)
- In 3 years the account will have \(\boxed{2653.02}\)