Answer :
To determine the year in which the price of 9-volt batteries will reach \$4 using the function \(P(t) = 1.1 \cdot e^{0.047t}\), follow these steps:
1. Set up the equation for the price goal:
[tex]\[ P(t) = 4 \][/tex]
Using the given function, substitute \(P(t)\):
[tex]\[ 1.1 \cdot e^{0.047t} = 4 \][/tex]
2. Solve for \(t\):
Divide both sides of the equation by 1.1:
[tex]\[ e^{0.047t} = \frac{4}{1.1} \][/tex]
3. Simplify the expression:
Calculate the right side:
[tex]\[ e^{0.047t} \approx 3.636 \][/tex]
4. Take the natural logarithm (ln) of both sides:
[tex]\[ \ln(e^{0.047t}) = \ln(3.636) \][/tex]
Using properties of logarithms:
[tex]\[ 0.047t = \ln(3.636) \][/tex]
5. Solve for \(t\):
[tex]\[ t = \frac{\ln(3.636)}{0.047} \][/tex]
Performing this calculation simplifies to:
[tex]\[ t \approx 27.4677485386290 \][/tex]
6. Determine the year:
Since \(t\) represents the number of years after 1980:
[tex]\[ \text{Year} = 1980 + t \approx 1980 + 27.4677485386290 \approx 2007 \][/tex]
Therefore, the price of 9-volt batteries will reach \$4 in the year 2007.
Thus, the correct answer is:
C. 2007
1. Set up the equation for the price goal:
[tex]\[ P(t) = 4 \][/tex]
Using the given function, substitute \(P(t)\):
[tex]\[ 1.1 \cdot e^{0.047t} = 4 \][/tex]
2. Solve for \(t\):
Divide both sides of the equation by 1.1:
[tex]\[ e^{0.047t} = \frac{4}{1.1} \][/tex]
3. Simplify the expression:
Calculate the right side:
[tex]\[ e^{0.047t} \approx 3.636 \][/tex]
4. Take the natural logarithm (ln) of both sides:
[tex]\[ \ln(e^{0.047t}) = \ln(3.636) \][/tex]
Using properties of logarithms:
[tex]\[ 0.047t = \ln(3.636) \][/tex]
5. Solve for \(t\):
[tex]\[ t = \frac{\ln(3.636)}{0.047} \][/tex]
Performing this calculation simplifies to:
[tex]\[ t \approx 27.4677485386290 \][/tex]
6. Determine the year:
Since \(t\) represents the number of years after 1980:
[tex]\[ \text{Year} = 1980 + t \approx 1980 + 27.4677485386290 \approx 2007 \][/tex]
Therefore, the price of 9-volt batteries will reach \$4 in the year 2007.
Thus, the correct answer is:
C. 2007
