Question
In 1979, the price of electricity was $0.05 per kilowatt-hour. The price of electricity has increased at a rate of approximately 2.05% annually. If t is the number of years after 1979, create the equation that can be used to determine how many years it will take for the price per kilowatt-hour to reach $0.10. Fill in the values of A, b, and c for this situation. Do not include dollar signs in the response.
Answers
Answered by
GPT-4o mini
To create the equation for the price of electricity over time, we can use the formula for exponential growth:
\[ P(t) = P_0 \cdot (1 + r)^t \]
Where:
- \( P(t) \) is the price at time \( t \),
- \( P_0 \) is the initial price,
- \( r \) is the growth rate,
- \( t \) is the number of years after 1979.
In this case:
- \( P_0 = 0.05 \) (the initial price in 1979),
- \( r = 0.0205 \) (the growth rate of 2.05% expressed as a decimal),
- We want to find the value of \( t \) when \( P(t) = 0.10 \).
Thus, the equation becomes:
\[ 0.10 = 0.05 \cdot (1 + 0.0205)^t \]
Now we can identify the values of \( A \), \( b \), and \( c \) as follows:
- \( A = 0.05 \)
- \( b = 1 + 0.0205 = 1.0205 \)
- \( c = 0.10 \)
So, the filled values are:
- A = 0.05
- b = 1.0205
- c = 0.10
\[ P(t) = P_0 \cdot (1 + r)^t \]
Where:
- \( P(t) \) is the price at time \( t \),
- \( P_0 \) is the initial price,
- \( r \) is the growth rate,
- \( t \) is the number of years after 1979.
In this case:
- \( P_0 = 0.05 \) (the initial price in 1979),
- \( r = 0.0205 \) (the growth rate of 2.05% expressed as a decimal),
- We want to find the value of \( t \) when \( P(t) = 0.10 \).
Thus, the equation becomes:
\[ 0.10 = 0.05 \cdot (1 + 0.0205)^t \]
Now we can identify the values of \( A \), \( b \), and \( c \) as follows:
- \( A = 0.05 \)
- \( b = 1 + 0.0205 = 1.0205 \)
- \( c = 0.10 \)
So, the filled values are:
- A = 0.05
- b = 1.0205
- c = 0.10