use the model a = a₀eᵏᵗ or ln(a/a₀)=kt. how long does it take for a deposit of $900 to double at 2%…

Explanation:

Step1: Identify the values of $A$, $A_0$ and $k$

We know that $A_0 = 900$, when the deposit doubles $A= 2\times900 = 1800$, and the interest rate $k = 0.02$.

Step2: Substitute into the continuous - compounding formula

The continuous - compounding formula is $A = A_0e^{kt}$, substituting the values we get $1800=900e^{0.02t}$.

Step3: Simplify the equation

Divide both sides of the equation $1800 = 900e^{0.02t}$ by $900$, we have $\frac{1800}{900}=e^{0.02t}$, which simplifies to $2 = e^{0.02t}$.

Step4: Take the natural logarithm of both sides

Taking the natural logarithm of both sides, $\ln(2)=\ln(e^{0.02t})$. Since $\ln(e^{x}) = x$, the right - hand side simplifies to $0.02t$. So, $\ln(2)=0.02t$.

Step5: Solve for $t$

We know that $\ln(2)\approx0.693147$. Then $t=\frac{\ln(2)}{0.02}=\frac{0.693147}{0.02}=34.65735$ years.

Step6: Convert the decimal part of years to days

The decimal part is $0.65735$. Since there are 365 days in a year, the number of days is $0.65735\times365\approx 240$ days.

Answer:

34 years 240 days