Question

In 1992, South Dakota's population was 10 million. Since then, the population has grown by 1.4% each year. Based on this, when will the population reach 20 million?

Answer 1

General form of an exponential equation:
`y=ab^x`

where:

• a is initial value
• b is the base (or growth factor in decimal form)
• x is the independent variable
• y is the dependent variable
• If b > 1 then it is an increasing function
• If 0 < b < 1 then it is a decreasing function
• Also b ≠ 0

Given information:

• initial population = 10 million
• growth rate = 1.4% each year

⇒ growth factor = 100% + 1.4% = 101.4% = 1.014

Inputting these values into the equation:

`\implies y=10(1.014)^x`

where y is the population (in millions) and x is the number of years since 1992

Now all we need to do is set y = 20 and solve for x:

`\implies 10(1.014)^x=20`

`\implies 1.014^x=2`

`\implies \ln 1.014^x=\ln 2`

`\implies x\ln 1.014=\ln 2`

`\implies x=(\ln 2)/(\ln 1.014)`

`\implies x=49.85628343...`

1992 + x = 2041.8562....

Therefore, the population will reach 20 million during 2041, so the population will reach 20 million by 2042.

Answer 2

• 49.9 years or 50 years later.

• At year : 2042

Step-by-step explanation:

use compound interest formula:
`\sf \boxed{ \sf P ( \sf 1 + (r)/(100) )^n}`

`\rightarrow \s \sf 10( 1 + (1.4)/(100) )^n = 20`

`\rightarrow \sf ( 1.014) ^n = 2`

`\rightarrow \sf n( ln( 1.014) ) = ln(2)`

`\rightarrow\sf n = (ln(2))/(ln( 1.014))`

`\rightarrow\sf n = 49.8563 \ years`