In the next 30 years traveling in a car is going look very d

In the next 30 years, traveling in a car is going look very different than it does today! Experts are predicting exponential growth both in the percent of cars that are shared among multiple users and in the percent of cars that are driverless. In 2020, about 2.6% of the total number of cars worldwide will be shared (for example, by Uber users); this percentage is forecast to growth by 19.2% per year. In 2020, driverless cars are predicted to make up about 0.8% of the total number of cars worldwide; this percentage is expected to triple every five years.

-Find formulas for the percentage of total cars on the road that are shared, S(t), and the percentage of total cars on the road that are driverless, D(t), as functions of time, in years since 2020.

-How long will it take for the percentage of total cars that are shared to triple?

-For both the percentage of total cars that are shared and that are driverless, find their continuous annuel growth rates, k.

Solution

-Find formulas for the percentage of total cars on the road that are shared, S(t), and the percentage of total cars on the road that are driverless, D(t), as functions of time, in years since 2020

Solution : Cars on road : 2020 , 2.6% cars ; growth rate = 19.2% = 0.192

S(t) = a(1+r)^t where a is the % of cars in 2020

=2.6( 1+0.192)^t

S(t) = 2.6(1.192)^t

Driveless cars:

0.8% of the total number of cars worldwide; this percentage is expected to triple every five years

D(t) = 0.8(1 +r)^t

0.8*3 = 0.8(1+r)^5---> 3 = (1+r)^5

1.246 = 1+r ----> r = =0.246 ---> r = 24.6%

Growth every year : r = 24.6%

D(t) = 0.8(1.246)^t

-How long will it take for the percentage of total cars that are shared to triple?

Solution:

S(t) = 2.6(1.192)^t

3*2.6 = 2.6(1.92)^t --->3 = (1.92)^t

taking log of both sides: ln3 = t*ln1.192

t = 6.25 years

-For both the percentage of total cars that are shared and that are driverless, find their continuous annuel growth rates, k.

Shared cars :continous grwoth factor k = 0.192

Driverless cars :continous grwoth factor k = 0.246