Answer :
Let's solve the problem step by step.
1. Understanding the Problem:
- We need to find the number of years it will take for the electricity consumption to triple given a continuous growth rate of 6%.
2. Using the Continuous Compound Growth Formula:
- The formula to model continuous growth is: [tex]\( A = P \cdot e^{(rt)} \)[/tex]
- [tex]\( A \)[/tex] is the final amount of electricity consumption.
- [tex]\( P \)[/tex] is the initial amount of electricity consumption.
- [tex]\( r \)[/tex] is the continuous growth rate.
- [tex]\( t \)[/tex] is the time in years.
- Here, we want [tex]\( A \)[/tex] to be three times [tex]\( P \)[/tex], so [tex]\( A = 3P \)[/tex].
3. Rearranging the Formula to Solve for Time [tex]\( t \)[/tex]:
- Start with [tex]\( 3P = P \cdot e^{(0.06t)} \)[/tex].
- Divide both sides by [tex]\( P \)[/tex] to get: [tex]\( 3 = e^{(0.06t)} \)[/tex].
- To solve for [tex]\( t \)[/tex], take the natural logarithm (ln) of both sides: [tex]\( \ln(3) = 0.06t \)[/tex].
- Isolate [tex]\( t \)[/tex]: [tex]\( t = \frac{\ln(3)}{0.06} \)[/tex].
4. Calculating [tex]\( t \)[/tex]:
- The natural logarithm of 3 ([tex]\( \ln(3) \)[/tex]) is approximately 1.0986.
- Therefore, [tex]\( t = \frac{1.0986}{0.06} \approx 18.310204811135165 \)[/tex].
5. Rounding the Result:
- To find the number of years to the nearest tenth, we round [tex]\( 18.310204811135165 \)[/tex] to 18.3.
So, the number of years before three times as much electricity would be needed is approximately 18.3 years.
1. Understanding the Problem:
- We need to find the number of years it will take for the electricity consumption to triple given a continuous growth rate of 6%.
2. Using the Continuous Compound Growth Formula:
- The formula to model continuous growth is: [tex]\( A = P \cdot e^{(rt)} \)[/tex]
- [tex]\( A \)[/tex] is the final amount of electricity consumption.
- [tex]\( P \)[/tex] is the initial amount of electricity consumption.
- [tex]\( r \)[/tex] is the continuous growth rate.
- [tex]\( t \)[/tex] is the time in years.
- Here, we want [tex]\( A \)[/tex] to be three times [tex]\( P \)[/tex], so [tex]\( A = 3P \)[/tex].
3. Rearranging the Formula to Solve for Time [tex]\( t \)[/tex]:
- Start with [tex]\( 3P = P \cdot e^{(0.06t)} \)[/tex].
- Divide both sides by [tex]\( P \)[/tex] to get: [tex]\( 3 = e^{(0.06t)} \)[/tex].
- To solve for [tex]\( t \)[/tex], take the natural logarithm (ln) of both sides: [tex]\( \ln(3) = 0.06t \)[/tex].
- Isolate [tex]\( t \)[/tex]: [tex]\( t = \frac{\ln(3)}{0.06} \)[/tex].
4. Calculating [tex]\( t \)[/tex]:
- The natural logarithm of 3 ([tex]\( \ln(3) \)[/tex]) is approximately 1.0986.
- Therefore, [tex]\( t = \frac{1.0986}{0.06} \approx 18.310204811135165 \)[/tex].
5. Rounding the Result:
- To find the number of years to the nearest tenth, we round [tex]\( 18.310204811135165 \)[/tex] to 18.3.
So, the number of years before three times as much electricity would be needed is approximately 18.3 years.
