Question

t, in years, since he invested the money, and the total amount of money B(t) is modeled by the following function:
b (t) = 2500 (21/20)^t



What factor does the balance increase by every decade?

Group of answer choices

Every decade the balance increases by a factor of 0.63.

Every decade the balance increases by a factor of 1.6

Every decade the balance increases by a factor of 0.05.

Every decade the balance increases by a factor of 1.05.

Answer 1

To determine the balance increase factor every decade for the function \( B(t) = 2500 \left(\frac{21}{20}\right)^t \), we can evaluate the function at \( t = 10 \) (for one decade) and see how it compares to the initial amount \( B(0) \).

1. Calculate \( B(0) \):
\[
B(0) = 2500 \left(\frac{21}{20}\right)^0 = 2500
\]

2. Calculate \( B(10) \):
\[
B(10) = 2500 \left(\frac{21}{20}\right)^{10}
\]

3. Now we need to find the factor of increase from \( B(0) \) to \( B(10) \):
\[
\text{Increase Factor} = \frac{B(10)}{B(0)} = \frac{2500 \left(\frac{21}{20}\right)^{10}}{2500} = \left(\frac{21}{20}\right)^{10}
\]

4. Calculate \( \left(\frac{21}{20}\right)^{10} \):
\[
\frac{21}{20} = 1.05
\]
So,
\[
\left(\frac{21}{20}\right)^{10} = (1.05)^{10}
\]

5. We can evaluate \( (1.05)^{10} \):
\[
(1.05)^{10} \approx 1.62889 \text{ (using a calculator)}
\]
which indicates the balance increases by a factor of approximately 1.63 every decade.

Therefore, the correct option should indicate that the balance increases by a factor of approximately 1.63 every decade. However, since that specific factor is not present in your answer choices, the closest one is **"Every decade the balance increases by a factor of 1.6."** Thus, the answer is:

**Every decade the balance increases by a factor of 1.6.**