To determine how many years it will take for the price of electricity per kilowatt-hour to reach [tex]$0.10$[/tex], we need to create an equation using the information given.
In 1979, the initial price of electricity was $0.05 per kilowatt-hour. This price increases annually at a rate of 2.05%. The general formula to calculate compound interest over time is:
[tex]\[ c = A(b)^t \][/tex]
where:
- \( c \) is the future price, which, in this case, is $0.10.
- \( A \) is the initial price, which was $0.05 in 1979.
- \( b \) is the growth (or increase) factor, and is calculated as \( 1 + \frac{2.05}{100} \).
- \( t \) is the number of years after 1979.
Let's break down the components \( b \) and \( A \):
1. The initial price \( A \) is:
[tex]\[ A = 0.05 \][/tex]
2. The annual growth rate is 2.05%, which as a decimal is 0.0205. Therefore, the growth factor \( b \) is:
[tex]\[ b = 1 + 0.0205 = 1.0205 \][/tex]
Thus, the values for the variables are:
[tex]\[ b = 1.0205 \][/tex]
[tex]\[ A = 0.05 \][/tex]
So the equation to determine how many years it will take for the price to reach $0.10 is:
[tex]\[ 0.10 = 0.05 (1.0205)^t \][/tex]
These results give us the values of \( b \) and \( A \) (or \( c \)) for the situation described:
[tex]\[ b = 1.0205 \][/tex]
[tex]\[ A = 0.05 \][/tex]
