Question

Maria studied the traffic trends in India. She found that the number of cars on the road increases by 10% each year. If there were 80 million cars in year 1 of her study, how many more cars were on the road in year 3 compared to year 2?

Answer 1

Explanation:

Let's first find the exponential function that models the situation in year one. The exponential standard form is

`y=a(b)^x` where a is the initial value and b is the growth/decay rate in decimal form. If it is growth it is added to 100% of the initial value; if it is decay it is taken away from 100% of the initial value. We are told that the number of cars in year one was 80 million, so

a = 80 (in millions)

If b is increasing by 10%, then we are adding that amount to the initial 100% we started with to give us 100% + 10% = 110% or, in decimal form, 1.1

The model for our situation is

`y=80(1.1)^x` where y is the number of cars after x years goes by. We want to find the difference between years 3 and 2, so we will use our model twice, replacing x with both a 2 and then a 3 and subtracting.

When x = 2:

`y=80(1.1)^2` and

y = 80(1.21) so

y = 96.8 million cars

When x = 3:

`y=80(1.1)^3` and

y = 80(1.331) so

y = 106.48 million cars

The difference between years 3 and 2 is

106.48 - 96.8 = 9.68 million cars