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

Answer:

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

Step-by-step 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:

$\frac{dC}{dt} = kC$

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

$C(t) = 4.4e^{kt}$

$5.5 = 4.4e^{5k}$

$e^{5k} = \frac{5.5}{4.4}$

$e^{5k} = 1.25$

$\ln{e^{5k}} = \ln{1.25}$

$5k = \ln{1.25}$

$k = \frac{\ln{1.25}}{5}$

$k = 0.0446$

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.