Question

San Diego is one of the largest cities in the United States. In 2012, the population of San Diego was 1,309,000. By 2016, the population had grown to 1,375,000, and reached 1,386,000 in 2020. (d) If the current growth rate continues, when will the population of San Diego reach 1.5 million?current growth rate is 0.20% from 2016-2020

Answer 1

In this problem, we have an exponential function of the form

y=a(b)^x

where

y is the population of San Diego

x number of years since 2012

so

Part a

2012 represents x=0, y=1,309,000

a=1,309,000

For the year 2016 ------> x=2016-2012=4 years

y=1,375,000

therefore

`1,375,000=1,309,000(b)^4`

solve for b

b=1.0124

b=1+r

substitute and solve for r

r=1.0124-1

r=0.0124

percentage

r=1.24%

Part b

2016-2020

in this part

a=1,375,000

x is the number of years since 2016

For the year 2020 -----> x=2020-2016=4 years

y=1,386,000

substitute

`1,386,000=1,375,000(b)^4`

solve for b

b=1.0020

b=1+r

r=1.0020-1

r=0.0020

percentage

r=0.20%

Part c

Compare the growth rate each period

2012-2016 ------> r=1.24%

2016-2020 -----> r=0.20%

that means ----> in the second period the growth rate has decreased than the first period

Part d

the current growth rate is r=0.20%

so

the equation is

`y=1,375,000(1.002)^x`

where

x is the number of years since 2016

so

For y=1,500,000

substitute

`1,500,000=1,375,000(1.002)^x`

solve for x

`(1,500,000)/(1,375,000)=(1.002)^x`

apply log on both sides

`\log ((1,500,000)/(1,375,000))=x\cdot\log (1.002)`

x=43.5 years

therefore

the year is

2016+43.5=2059.5 -------> approximate the year 2060