Question

I need part b but part a would also be helpful​

Answer 1

`\textsf{a)}\quad P(t)=310000(1.0085)^t`

where P is the population, and t is the number of years after 500 BC.

b) 42,013,000

Explanation:

`\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}`

Define the variables:

• P = population
• t = time (in years)

The population in 500 BC is the initial value (when t = 0), therefore if the population was 310,000 in 500 BC:

• a = 310000

If the population increases by 0.85% each year then the growth rate in decimal form is:

• b = 1.0085

Therefore, the function that models the given scenario is:

`\boxed{P(t)=310000(1.0085)^t}`

where P is the population, and t is the number of years after 500 BC.

To use the function to calculate the population in 80 AD, first calculate the number of years (value of t):

⇒ 80 AD - 500 BC = 580 years

Therefore, to find the population in 80 AD, substitute t = 580 into the function:

`\implies P(580)=310000(1.0085)^(580)`

`\implies P(580)=42013142.9844236...`

Therefore, the population in 80 AD is 42,013,000 (to the nearest thousand).