91Ó°ÊÓ

Do you see that \(\left(1+\frac{0.065}{365}\right)^{365 t}\) looks a lot like \( \lim_{{n \to \infty}} \left(1 + \frac{1}{n}\right)^n \) which is equal to \( e \)? Let's use that fact to simplify the equation.\nFirst, rewrite the base the equation so that it mirrors the definition of Euler's Number (e). We then split up the exponent \( 365 \cdot t \) into \( 365 \cdot \frac{0.065}{365} \cdot t \), which leads to the equation \( \left(1+\frac{0.065}{365}\right)^{\frac{0.065}{0.065} \cdot t}=4 \). Now we can see that the inside of the parentheses will approach one. Thus, the expression \( \left(1+\frac{0.065}{365}\right)^{365} \) tends towards \( e^{0.065} \) Therefore, the equation can be rewritten as \( e^{0.065 \cdot t} = 4 \)

We now have the expression in a much more convenient form. We can apply the natural log (ln) to both sides of our equation \( e^{0.065 \cdot t} = 4 .\)The special property of \( ln( e^x) = x \) simplifies the left side of the equation. We hence obtain \( 0.065 \cdot t = ln(4) \).

Now you can solve for \( t \). Achieve this by dividing ln(4) by 0.065. After approximating this to three decimal places (approx t = ln(4)/0.065 ≈ 21.199).

Lastly, check if the obtained value for \( t \) is correct by substituting it into the original equation and checking if both sides are equal. It should return a truth value close to true Due to the decimal approximation, there won't be an exact equality.