a) The annual inflation rate is 3.5% per year. If a movie ticket costs $7.50, find a formula for p, the price of the tickets t years from today, assuming that movie tickets keep up with inflation.
b) According to your formula, how much will movie tickets cost in 20 years?
a) To find a formula for the price of movie tickets t years from today, assuming they keep up with inflation, we can use the formula for compound interest.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the present value of the investment (in this case, the starting ticket price)
r = the annual interest rate (in this case, the inflation rate)
n = the number of compounding periods per year (since inflation is typically calculated annually, n = 1)
t = the number of years
In this case, the starting ticket price (P) is $7.50 and the inflation rate (r) is 3.5% per year.
Therefore, the formula for the price of tickets t years from today is:
p = 7.50(1 + 0.035/1)^(1t)
p = 7.50(1 + 0.035)^t
p = 7.50(1.035)^t
b) To calculate how much movie tickets will cost in 20 years, we can substitute the value of t = 20 into the formula:
p = 7.50(1.035)^20
p ≈ 12.15
Therefore, according to the formula, movie tickets are estimated to cost around $12.15 in 20 years.
a) To find a formula for the price of movie tickets t years from today, assuming they keep up with inflation, we need to consider the annual inflation rate.
The formula can be written as p = c * (1 + r)^t, where p is the price of the tickets t years from today, c is the current price of the tickets, r is the annual inflation rate expressed as a decimal, and t is the number of years.
In this case, the current price of the tickets is $7.50, and the annual inflation rate is 3.5% per year. Converting the inflation rate to a decimal, we get r = 0.035.
Therefore, the formula for p, the price of the tickets t years from today, is p = 7.50 * (1 + 0.035)^t.
b) To calculate how much movie tickets will cost in 20 years, we substitute t = 20 into the formula we derived earlier.
p = 7.50 * (1 + 0.035)^20
Calculating this value will give us the estimated price of movie tickets in 20 years, assuming they keep up with the specified inflation rate.
p(t) = 7.50 * 1.035^t
p(20) = 14.92