Question

1. a parent invests $2,000 at a 5% interest rate to help his daughter save for college. how many years will it take his money to double? round to the nearest thousandth.

Explanation:

Step1: Use the compound - interest formula

The compound - interest formula is \(A = P(1 + r)^{t}\), where \(A\) is the final amount, \(P\) is the principal amount, \(r\) is the annual interest rate (in decimal form), and \(t\) is the number of years.
We know that \(P=\$2000\), \(r = 0.05\), and \(A = 2P=\$4000\). Substitute these values into the formula: \(4000=2000(1 + 0.05)^{t}\).

Step2: Simplify the equation

Divide both sides of the equation \(4000 = 2000(1.05)^{t}\) by \(2000\). We get \(2=(1.05)^{t}\).

Step3: Take the natural logarithm of both sides

Using the property \(\ln a^{b}=b\ln a\), if \(2=(1.05)^{t}\), then \(\ln 2=t\ln(1.05)\).

Step4: Solve for \(t\)

We know that \(\ln 2\approx0.6931\) and \(\ln(1.05)\approx0.0488\). Then \(t=\frac{\ln 2}{\ln(1.05)}\).
Substitute the values: \(t=\frac{0.6931}{0.0488}\approx14.203\).

Answer:

\(t\approx14.203\) years.