Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar.

david invested $220 in a savings account that offers a 3% return on the investment. the value of davids investment will be at least $400 after a period of \boxed{} years.

hint: use the formula $a = p(1 + r)^t$, where $a$ is the amount after $t$ years, $p$ is the amount invested, $r$ is the rate of interest, and $t$ is the time period. use a calculator to compute the answer, and round it off to the nearest year.

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Explanation:

Step1: Identify given values

We know that \( P = 220 \), \( r = 0.03 \) (since 3% = 0.03), and \( A = 400 \). The formula is \( A = P(1 + r)^t \).

Step2: Substitute values into formula

Substitute the known values into the formula: \( 400 = 220(1 + 0.03)^t \).

Step3: Simplify the equation

First, divide both sides by 220: \( \frac{400}{220}=(1.03)^t \). Simplify \( \frac{400}{220} \) to \( \frac{20}{11}\approx1.8182 \). So we have \( 1.8182=(1.03)^t \).

Step4: Take natural logarithm of both sides

Take \( \ln \) of both sides: \( \ln(1.8182)=t\ln(1.03) \).

Step5: Solve for t

We can solve for \( t \) by dividing both sides by \( \ln(1.03) \): \( t=\frac{\ln(1.8182)}{\ln(1.03)} \).

Step6: Calculate the value

Using a calculator, \( \ln(1.8182)\approx0.5978 \) and \( \ln(1.03)\approx0.0296 \). Then \( t=\frac{0.5978}{0.0296}\approx20.2 \). Rounding to the nearest year, \( t\approx20 \).

Answer:

20