Question

In 1963, the number of cars in the U.S. was about 1.7 million. The number of cars grows at about 2.2% per year. Write an exponential equation to model this situation. Next find the number of cars in the year 1979 (round to one decimal place). Finally find out what year (round to the nearest year) it would have been when the number of cars reached 2.9 million. Show all work.

Answer 1

To model the situation of the number of cars growing at about 2.2% per year, we can use the exponential equation:

N(t) = N₀ * (1 + r)^t

Where:
N(t) is the number of cars at time t,
N₀ is the initial number of cars,
r is the growth rate expressed as a decimal,
t is the number of years.

Given:
N₀ = 1.7 million,
r = 2.2% = 0.022.

1) Finding the number of cars in the year 1979:
To find the number of cars in a specific year, we substitute the value of t with the number of years from the initial year (1963) to the target year (1979).

t = 1979 - 1963 = 16 years

N(16) = 1.7 million * (1 + 0.022)^16

Calculating this value, we find that the number of cars in 1979 was approximately 3.45 million (rounded to one decimal place).

2) Finding the year when the number of cars reached 2.9 million:
To find the year, we rearrange the equation:

2.9 million = 1.7 million * (1 + 0.022)^t

Dividing both sides by 1.7 million:

2.9/1.7 = (1 + 0.022)^t

Using logarithms, we can solve for t:

t = log(2.9/1.7) / log(1 + 0.022)

Calculating this value, we find that t is approximately 19.4 years.

Therefore, the year when the number of cars reached 2.9 million would be approximately 1982 (rounded to the nearest year).