In 1979, the price of electricity was [tex]$\$[/tex] 0.05[tex]$ per kilowatt-hour. The price of electricity has increased at a rate of approximately $[/tex]2.05\%[tex]$ annually. If $[/tex]t[tex]$ 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 $[/tex]\[tex]$ 0.10$[/tex].

Fill in the values of [tex]$b$[/tex] and [tex]$c$[/tex] for this situation. Do not include dollar signs in the response.

The equation is:
[tex]\[ c = A(b)^t \][/tex]

(Note: Disregard any unrelated or erroneous content from the original text.)



Answer :

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]