Question

The annual energy consumption of the town where Camilla lives in creases at a rate that is onal at any time to the energy consumption at that time. The town consumed 4.4 trillion ually after 5 years. British thermal units (BTUs) initially, and it consumed 5.5 trillion BTUs ann What is the town's annual energy consumption, in trillionso f BTUs, after 9 years?

Answer 1

6.575 trillion BTUs

Explanation:

Let represent the annual energy consumption of the town as E

The rate of annual energy consumption * energy consumption at time past

dE/dt * E

dE/dt =K

k = the proportionality constant

c= the integration constant

(dE/dt=) kdt

lnE = kt + c

E(t) = e^kt+c ⇒ e^c e^kt e^c is a constant, and e^c = E₀

E(t) = E₀ e^kt

The initial consumption of energy is E(0)=4.4TBTU

set t = 0 then

4.4 = E₀ e⇒ E₀ (1)

E₀ = 4.4

E (t) = 4.4e^kt

The consumption after 5 years is t = 5, e(5) = 5.5TBTU

so,

E(5) = 5.5 = 4.4e^k(5)

e^5k = 5/4

We now take the log 5kln = ln(5/4)

5k(1) = ln(5/4)

k = 1/5 ln(5/4) = 0.04463

We find the town's annual energy consumption, after 9 years

we set t=9

E(9) = 4.4e^0.04463(9)

= 4.4(1.494301) = 6.5749TBTUs

Therefore the annual energy consumption of the town after 9 years is

= 6.575 trillion BTUs