attempt 1: 10 attempts remaining. the population of south carolina can be modeled by p(t) million people, t…

attempt 1: 10 attempts remaining. the population of south carolina can be modeled by p(t) million people, t years after 1990. the number of registered republicans in south carolina is given by r(p) million republicans when the population of south carolina is p million people. p(5)=4.527, dp/dt|t = 5=0.078, r(4.527)=1.142, dr/dp|p = 4.527=0.43 a. how many registered republicans were there in south carolina in 1995? ? million republicans b. in 1995, when the population was 4.527 million people, how quickly was the number of registered republicans in south carolina changing with respect to the size of the population? ? million republicans per million people c. in 1995, how quickly was the number of registered republicans in south carolina changing with respect to time? ? million republicans per year

attempt 1: 10 attempts remaining. the population of south carolina can be modeled by p(t) million people, t years after 1990. the number of registered republicans in south carolina is given by r(p) million republicans when the population of south carolina is p million people. p(5)=4.527, dp/dt|t = 5=0.078, r(4.527)=1.142, dr/dp|p = 4.527=0.43 a. how many registered republicans were there in south carolina in 1995? ? million republicans b. in 1995, when the population was 4.527 million people, how quickly was the number of registered republicans in south carolina changing with respect to the size of the population? ? million republicans per million people c. in 1995, how quickly was the number of registered republicans in south carolina changing with respect to time? ? million republicans per year

Answer

Explanation:

Step1: Find number of registered Republicans in 1995

In 1995, (t = 5). We know that (r(p(5))) gives the number of registered Republicans. Given (p(5)=4.527) and (r(4.527) = 1.142), so the number of registered Republicans in 1995 is (r(p(5))=1.142).

Step2: Rate of change of Republicans with respect to population

The rate of change of the number of registered Republicans with respect to the size of the population is given by (\frac{dr}{dp}). When (p = 4.527), (\frac{dr}{dp}\big|_{p = 4.527}=0.43).

Step3: Rate of change of Republicans with respect to time

By the chain - rule, (\frac{dr}{dt}=\frac{dr}{dp}\cdot\frac{dp}{dt}). We know that (\frac{dr}{dp}\big|{p = 4.527}=0.43) and (\frac{dp}{dt}\big|{t = 5}=0.078). So (\frac{dr}{dt}\big|_{t = 5}=0.43\times0.078 = 0.03354).

Answer:

a. 1.142 b. 0.43 c. 0.03354