How long will it take $1900 invested at an 9% compounded monthly to double? Round to two decimal places.
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2 = (1 + r/n)^(nt) Show more…
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Continuously Compounded Interest Use the following discussion: Suppose an initial investment, called the principal $P,$ earns an annual rate of interest $r,$ which is compounded $n$ times per year. The interest earned on the principal $P$ in the first compounding period is $P\left(\frac{r}{n}\right)$, and the resulting amount A of the investment after one compounding period is $A=P+P\left(\frac{r}{n}\right)=P\left(1+\frac{r}{n}\right) .$ After $k$ compounding periods, the amount $A$ of the investment is $A=P\left(1+\frac{r}{n}\right)^{k}$. Since in $t$ years there are nt compounding periods, the amount A after t years is $$ A=P\left(1+\frac{r}{n}\right)^{n t} $$ When interest is compounded so that after $t$ years the accumulated amount is $A=\lim _{n \rightarrow \infty} P\left(1+\frac{r}{n}\right)^{n t},$ the interest is said to be compounded continuously. (a) Show that if the annual rate of interest $r$ is compounded continuously, then the amount $A$ after $t$ years is $A=P e^{r t}$, where $P$ is the initial investment. (b) If an initial investment of $P=\$ 5000$ earns $2 \%$ interest compounded continuously, how much is the investment worth after 10 years? (c) How long does it take an investment of $\$ 10,000$ to double if it is invested at $2.4 \%$ compounded continuously? (d) Show that the rate of change of $A$ with respect to $t$ when the interest rate $r$ is compounded continuously is $\frac{d A}{d t}=r A$.
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Adi S.
Suppose that P₀ is invested in a savings account in which interest is compounded continuously at 6.5% per year. That is, the balance P grows at the rate given by the following equation. dP/dt = 0.065P(t) (a)Find the function P(t) that satisfies the equation. Write it in terms of P₀ and 0.065. (b)Suppose that $1500 is invested. What is the balance after 2 years? (c)When will an investment of $1500 double itself? (a) Choose the correct answer below. A. P(t) = P(t) e^0.065t B. P₀ = P(t) e^0.065t C. P(t) = P₀ e^0.065t D. P(t) = 0.065P₀ e^t (b) The balance after 2 year is $. (Type an integer or decimal rounded to two decimal places as needed.) (c) The doubling time is year. (Type an integer or decimal rounded to two decimal places as needed.)
Madhur L.
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