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. $\newline$ The town consumed $4$ . $4$ trillion British thermal units (BTUs) initially, and it consumed $5$ . $5$ trillion BTUs annually after $5$ years. $\newline$ What is the town's annual energy consumption, in trillions of BTUs, after $9$ years? $\newline$ Choose $1$ answer: $\newline$ (A) $6$ . $6$ $\newline$ (B) $32$ . $8$ $\newline$ (C) $41$ . $8$

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Q. 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.

\newline

The town consumed

4

4

trillion British thermal units (BTUs) initially, and it consumed

5

5

trillion BTUs annually after

5

years.

\newline

What is the town's annual energy consumption, in trillions of BTUs, after

9

years?

\newline

Choose

1

answer:

\newline

(A)

6

6

\newline

(B)

32

8

\newline

(C)

41

8

Identify Initial and Final Amounts: Identify the initial and final amounts of energy consumption and the time it took to reach the final amount from the initial amount. $\newline$ Initial energy consumption: $4.4$ trillion BTUs $\newline$ Energy consumption after $5$ years: $5.5$ trillion BTUs $\newline$ Time taken to reach $5.5$ trillion BTUs from $4.4$ trillion BTUs: $5$ years
Calculate Rate of Increase: Calculate the rate of increase per year using the proportional relationship. $\newline$ Let the rate of increase be represented by ' extit{k}' (a constant). $\newline$ Using the formula for exponential growth: Final amount = Initial amount $\times e^{(k \times \text{time})}$ $\newline$ $5.5 = 4.4 \times e^{(k \times 5)}$
Solve for Rate Constant: Solve for the rate constant $k$ . $\frac{5.5}{4.4} = e^{k \times 5}$ $1.25 = e^{k \times 5}$ Take the natural logarithm (ln) of both sides to solve for $k$ . $\ln(1.25) = \ln(e^{k \times 5})$ $\ln(1.25) = k \times 5$ $k = \frac{\ln(1.25)}{5}$
Calculate Value of 'k': Calculate the value of 'k'. $\newline$ $k = \frac{\ln(1.25)}{5}$ $\newline$ $k \approx \frac{0.045}{5}$ $\newline$ $k \approx 0.009$
Use 'k' to Find Consumption: Use the rate constant 'k' to find the energy consumption after $9$ years. $\newline$ Energy consumption after $9$ years = Initial amount $\times e^{(k \times \text{time})}$ $\newline$ Energy consumption after $9$ years = $4.4 \times e^{(0.009 \times 9)}$
Calculate Consumption After $9$ Years: Calculate the energy consumption after $9$ years. $\newline$ Energy consumption after $9$ years = $4.4 \times e^{(0.009 \times 9)}$ $\newline$ Energy consumption after $9$ years $\approx 4.4 \times e^{0.081}$ $\newline$ Energy consumption after $9$ years $\approx 4.4 \times 1.08447$ $\newline$ Energy consumption after $9$ years $\approx 4.77167$ trillion BTUs
Round Final Answer: Round the final answer to one decimal place as the options are given in one decimal place. $\newline$ Energy consumption after $9$ years $\approx 4.8$ trillion BTUs