The population of South Carolina can be modeled by p)3.48(1.023\') million peopl

Question

The population of South Carolina can be modeled by p)3.48(1.023') million people, t years after 1990. The number of registered Republicans in South Carolina is given by r(p) 0.424p million people when the population of South Carolina is p million people a. How many registered Republicans were there in South Carolina in 2016? (2 pts) How quickly was the number of registered Republicans in South Carolina changing with respect to time in 2016? b. (2 pts) c. Complete the model for r(t) (2 pts) r(t) = firnction units gives the number of registered Republicans in South Carolina t years after 1990. d. Write a completely defined model for r'(t) (3 pts)

Explanation / Answer

First of all, population in 2016, that is, after 26 yrs after 1990 will be 3.48(1.023)^26 million. On simplification it works out to 6,285,585

a) The number of registered republicans in 2016 would be the number given by r(p)=0.424p = 0.424(6285585)=2,665,088 or 2.66 million.

b) Rate of change of republicans would be dr(p)/dt= 0.424 dp/dt . dp/dt can be had by differentiating p=3.48 (1.023^t). dp/dt= 3.48 (1.023^t) ln (1.023). Thus dr(p)/dt = 0.424 (3.48) (1.023^t) ln (1.023). For 2016, t=26, hence the rate of change of number of regeistered republicans would be 0.424(3.48)(1.023)^26 ln (1.023)=0.06 million per year

c) Since r(p)= 0.424 p, r(t)= 0.424 p(t)= 0.424 (3.48) (1.023)^t = 1.47552 (1.023)^t million

d) r'(t)= 1.47552 d/dt (1.023)^t = 1.47552 (1.023)^t ln (1.023)

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