math problems?

Question

2)How long it will take for an investment of 3000 dollars to double in value if the interest rate is 7 percent per year, compounded continuously?
Your answer is t=____ years.

3)For a pole-vaulter in training, the height the athelete will be able to pole-vault after t months of training is given by
H(t)=20-15e^-0.02t

After how many months will she be able to vault 15 ft?

_____________months

Answer 1

Continuous compounding suggests daily compounding:
A = Ao(1 + i/365)^(365y)
(1 + i/365)^(365y) = A/Ao
365yln(1 + 0.07/365) = ln(2)
y = ln(2)/[365ln(1.00019178)
y = 0.69314718/0.06999329
y ≈ 9.903052 years or
y ≈ 9 years 330 days

3) H(t) = 20 - 15e^-0.02t
20 - 15e^-0.02t = 15
4 - 3e^-0.02t = 3
3e^-0.02t = 1
e^-0.02t = 1/3
- 0.02t = ln(1/3)
t = ln(3) / 0.02
t = 54.93 months

Answer 2

Remember that the formula for continuous compounding is

A = Ao * e^(r*t), where
A = the current amount
Ao = the initial amount invested
r = the interest rate
t = the time in years

For you, A is supposed to be twice Ao, and you have r, so rearrange your equation to get t by itself and plug in your numbers.

The only other thing you need to remember to solve both problems is that ln (e^x) = x, and you can take the ln of both sides of an equation without affecting the relationship between the two sides.

The answer from Alexi L is completely wrong. It uses simple annual interest instead of continuously compounded interest.

Answer 3

2)

7 percent is equal to 7/100 = 0.07
For 3000 to be doubled, you have to add +3000 to it. Now let us multiply 3000 by 7/100 and see how much money we will get extra each year:

3000 * 7/100 = 3000 * 0.07 = 210

We are making 210 money each year. Now divide 3000 by 210 and see how many years it will take for us to make 3000 dollars:

3000 / 210 = 14.28

So it will take about 14 to 15 years for the money to be doubled up.


Good luck.