Question

If Sally invested $45,000 in an account that pays 11.9% interest, compounded continuously:

How much would be available after 10 years?

How much after 25 years?

How long will it take to double in value?

Answer 1

Answer:

Explanation:
The formula for continuously compounding interest is P(at the time t) = P(original) * [tex]e^{rt}[tex]

P(t) = value at time t
P(o)= original principal (amt invested)
r = nominal annual interest rate
t = length of time the interest is applied

r is expressed as a decimal, so 20% is 0.20, and in this example 11.9% would be 0.119.

After 10 years:
P(t) = 45,000 * e ^(0.119*10)
=$147,918.65

After 25 years:
P(t) = 45,000 * e ^(0.119*25)
= $881,533.05

To double in value, $45,000 x 2 = $90,000.
So the equation would be $90,000 = $45,000 * e^(0.119t)
Now solve for t.
90,000/45,000 = e^(0.119t)
2=e^(0.119t)
Take the log of both sides
log(2) = log(e^(0.119t))
The logarithm of a number raised to a power is the power times the logarithm of the number.
0.119t=log(2)/log(e)
0.119t= 0.693147
Now solve for t
t=5.82

You can go back to the original formula to check:
=45000*e^(0.119*5.82)
=90,000
log(2)
​

Answer 2

Answer: To solve this problem, we can use the formula for continuous compound interest:

A = Pe^(rt)

Explanation:

A = Pe^(rt)

where A is the amount of money in the account after t years, P is the principal amount (initial investment), e is the mathematical constant (approximately 2.71828), r is the annual interest rate as a decimal, and t is the time in years.

For Sally's investment:

P = $45,000

r = 0.119 (11.9% as a decimal)

a) After 10 years:

t = 10

A = 45000e^(0.119 x 10) = $158,086.34

Therefore, after 10 years, Sally would have approximately $158,086.34 in the account.

b) After 25 years:

t = 25

A = 45000e^(0.119 x 25) = $989,988.27

Therefore, after 25 years, Sally would have approximately $989,988.27 in the account.

c) To find how long it takes to double in value:

We can use the formula:

2P = Pe^(rt)

Simplifying this equation:

2 = e^(rt)

ln(2) = rt

t = ln(2)/r

Using the values of P and r from above:

t = ln(2)/0.119 = 5.81 years (rounded to two decimal places)

Therefore, it would take approximately 5.81 years for Sally's investment to double in value.