Question

consider that $15(1.055)^x$ shows the balance in a savings account over time with no additional deposits or withdrawals. which statement is the best interpretation of one of the values in this function? a the savings account earned 15% interest for the first year. b the savings account earns an annual rate of interest of 5.5%. c the savings account has earned interest for 15 years. d the starting balance of the savings account was $150.

Explanation:

Step1: Recall the compound interest formula

The general formula for compound interest (or exponential growth in finance) is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal (initial amount), $r$ is the annual interest rate (in decimal), and $t$ is the time in years.

Step2: Analyze the given function $15(1.055)^x$

• Comparing with $P(1 + r)^t$, we have:
• $P = 15$: This would be the initial balance (principal). But wait, option D says the starting balance is $150$, which is incorrect.
• $1 + r = 1.055$: Solving for $r$, we get $r = 1.055 - 1 = 0.055$, which is $5.5\%$ annual interest rate.
• $x$ is the time (in years, presumably).

Step3: Evaluate each option

• Option A: The interest rate is $5.5\%$, not $15\%$. So A is wrong.
• Option B: Since $1 + r = 1.055$ implies $r = 0.055 = 5.5\%$ annual rate, B is correct.
• Option C: The "15" is the principal, not the number of years. So C is wrong.
• Option D: The principal is $15$ (assuming units are, say, dollars), not $150$. So D is wrong.

Answer:

B. The savings account earns an annual rate of interest of 5.5%.