Question

The annual energy consumption of the town where Camilla lives increases at a rate that is proportional at any time to the energy consumption at that time. The town consumed 4.44.44, point, 4 trillion British thermal units (BTUs) initially, and it consumed 5.55.55, point, 5 trillion BTUs annually after 555 years. What is the town's annual energy consumption, in trillionso f BTUs, after 9 years?

Answer 1

The town's annual energy consumption will be of 6.57 trillons of BTU after 9 years.

Explanation:

The annual energy consumption of the town where Camilla lives increases at a rate that is proportional at any time to the energy consumption at that time.

This means that the consumption after t years is given by the following differential equation:

`(dC)/(dt) = kC`

In which k is the growth rate.

The solution is, applying separation of variables:

`C(t) = C(0)e^(kt)`

In which C(0) is the initial consumption.

The town consumed 4.4 trillion British thermal units (BTUs) initially.

This means that
`C(0) = 4.4`

So

`C(t) = C(0)e^(kt)`

`C(t) = 4.4e^(kt)`

5.5 trillion BTUs annually after 5 years.

This means that
`C(5) = 5.5`. We use this to find k. So

`C(t) = 4.4e^(kt)`

`5.5 = 4.4e^(5k)`

`e^(5k) = (5.5)/(4.4)`

`e^(5k) = 1.25`

`\ln{e^(5k)} = ln(1.25)`

`5k = ln(1.25)`

`k = (ln(1.25))/(5)`

`k = 0.0446`

So

`C(t) = 4.4e^(0.0446t)`

After 9 years?

This is C(9). So

`C(9) = 4.4e^(0.0446*9) = 6.57`

The town's annual energy consumption will be of 6.57 trillons of BTU after 9 years.