Question

a. Tell whether the model represents exponential growth or exponential decay.b. Identify the annual percent increase or decrease in population.c. Estimate when the population was about 590,000.

Answer 1

The form of the exponential function is

`y=a(1\pm r)^x`

a is the initial amount

r is the percent of increase or decrease in decimal

We use + when it is growth

We use - when it is decay

The given exponential function is

`P=494.29(1.03)^t`

By comparing it with the model above

a = 494.29 ---- initial population

1 + r = 1.03

a)

Since 1.03 is greater than 1, then

The model represents exponential growth

b)

Since 1 + r = 1.03, then subtract 1 from each side to find r

`\begin{gathered} 1+r=1.03 \\ 1-1+r=1.03-1 \\ r=0.03 \end{gathered}`

Change it to percent by multiplying it by 100%

`0.03*100\text{ \%=3\%}`

The annual percentage of increase is 3%

c)

If the population is 590,000, then substitute P by this value and solve to find t

`590000=494.29(1.03)^t`

Divide each side by 494.29

`(590000)/(494.29)=(1.03)^t`

Insert ln for both sides

`\ln ((590000)/(494.29))=\ln (1.03)^t`

Use the rule

`\ln (a)^b=b\ln (a)`
`\ln ((590000)/(494.29))=t\ln (1.03)`

Divide both sides by ln(1.03) to find t

`\begin{gathered} (\ln ((590000)/(494.29)))/(\ln (1.03))=t \\ 239.68344=t \end{gathered}`

Round it to the nearest years

t = 240

The population will be 590,000 after 240 years