Mathematics, 15.01.2020 04:31 bookS4169

Compound interest a deposite of $1000 is made in an account that earns interest at an annual rate of 5%.how long will it take for the balance to double when the interest is compounded (a) annually,(b) monthly,(c) daily, and (d) continously?

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Answer from: sweetgigi5950

(a) 14.21

(b) 13.78

(d) 13.86

Step-by-step explanation:

Compound Interest, compounded n times per year,

$A = P(1 + \frac{r}{n} )^{nt}$

Compound Interest, compounded continuously,

$A = Pe^{rt}$

Where

A is the new amount/balance

P is the Principal (amount invested)

r is the rate in percentage

n is the number of times it is compounded.

Given:

Principal, P = $1000

rate, r = 5% = 0.05

For the balance to double, A = 2 × $1000

A = $2000

(a) annually, n = 1

$2000 = 1000(1 + \frac{0.05}{1} )^{1 * t}$

$2 = (1 + 0.05)^{t}$

$2 = (1.05)^{t}$

To solve for t, we take $log_{10}$ of both sides

$log_{10}2 = log_{10}1.05^{t}$

$log_{10}2 = t(log_{10}1.05)$

$t = \frac{log_{10}2}{log_{10}1.05}$

$t = \frac{0.3010}{0.0212}$

t = 14.21

(b) monthly, n = 12

$2000 = 1000(1 + \frac{0.05}{12} )^{12 * t}$

$2 = (1 + 0.0042)^{12t}$

$2 = (1.0042)^{12t}$

To solve for t, we take $log_{10}$ of both sides

$log_{10}2 = log_{10}1.0042^{12t}$

$log_{10}2 = 12t(log_{10}1.0042)$

$t = \frac{log_{10}2}{12 * log_{10}1.0042}$

$t = \frac{0.3010}{0.0218}$

t = 13.78

$2000 = 1000(1 + \frac{0.05}{365} )^{365 * t}$

$2 = (1 + 0.00014)^{365t}$

$2 = (1.00014)^{365t}$

To solve for t, we take $log_{10}$ of both sides

$log_{10}2 = log_{10}1.00014^{365t}$

$log_{10}2 = 365t(log_{10}1.00014)$

$t = \frac{log_{10}2}{365 * log_{10}1.00014}$

$t = \frac{0.3010}{0.022}$

t = 13.565

t = 13.57

(d) Continuously

$2000 = 1000e^{0.05t}$

$2 = e^{0.05t}$

To solve for t, we take $log_{e}$ of both sides

$log_{e}2 = log_{e}e^{0.05t}$

But log_{e}e = 1

In2 = 0.05t

$t =\frac{In2}{0.05}$

$t =\frac{0.693}{0.05}$

t = 13.86

Answer from: Quest

B.-2,3 because it is touch the -3 i think that the answer

Answer from: Quest

The answer is harsh laws are needed to control the society

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