Question

you invest $4000 in a bank account with 5% interest compounded annually. you are wondering how long it would take for your money to double. what equation will you use to solve the problem and how long will it take for your money to double? a. 8000 = 4000(1.05)^t; 14.2 years b. 4000 = 8000(1.05)^t; 2 years c. 8000 = 4000(1.05)^t; 7 years d. 4000 = 8000(1.05)^t; 28 years please select the best answer from the choices provided a b c d

Explanation:

Step1: Recall Compound Interest Formula

The compound interest formula is $A = P(1 + r)^t$, where $A$ is the amount, $P$ is the principal, $r$ is the annual interest rate (decimal), and $t$ is time in years. Here, $P = 4000$, $r = 0.05$, and we want $A$ to double, so $A = 8000$.

Step2: Form the Equation

Substitute values into the formula: $8000 = 4000(1 + 0.05)^t$, which simplifies to $8000 = 4000(1.05)^t$.

Step3: Solve for \( t \)

Divide both sides by 4000: $2 = (1.05)^t$. Take the natural logarithm of both sides: $\ln(2) = t\ln(1.05)$. Then $t = \frac{\ln(2)}{\ln(1.05)} \approx 14.2$ years.

Step4: Match with Options

Option a has the equation $8000 = 4000(1.05)^t$ and time 14.2 years, which matches our derivation.

Answer:

A. \( 8000 = 4000(1.05)^t \); 14.2 years