# It is known that a capital `C_(0)` applied to fixed interest capitalized continuously grows at a rate proportional to the capital present at any instant. If the initial capital has doubled in 5...

It is known that a capital `C_(0)` applied to fixed interest capitalized continuously grows at a rate proportional to the capital present at any instant. If the initial capital has doubled in 5 years, when he will triple? As quadruple?

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tiburtius | Certified Educator

Formula for capital `C` at time `t` with fixed interest capitalized continuously that grows at rate proportional to the capital present at time `t` is

`C(t)= C_0 e^(rt)`                                                               (1)

where `C_0` is initial capital and `r` is interest rate.

You can prove formula (1) by solving differential equation

`C'(t)=rC(t)`

`C(0)=C_0`

Let's now solve your problem. Since initial capital is doubled after 5 years we have

`C(5)=2C_0`

`C_0e^(5r)=2C_0`

We can now calculate `r.`

`e^(5r)=2`

Now we take natural logarithm.

`5r=ln2`

`r=ln2/5`

Now that we have `r` we can calculate anything we want.

When will our initial capital triple?

`C_0 e^(ln2/5 t)=3C_0`

`e^(ln2/5t)=3`

`ln2/5t=ln3`

`t=(5ln3)/ln2approx7.92`

Initial capital will quadruple in 7.92 years.

When will our initial capital quadruple?

Since our capital is doubled every 5 years initial capital should quadruple in 10 years. Let's check that.

`C_0e^(ln2/5t)=4C_0`

`e^(ln2/5t)=4`

`ln2/5t=ln4`

`t=(5ln4)/ln2=5cdot2=10`

Initial capital will quadruple in 10 years.

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