Question

the population of toledo, ohio, in the year 2000 was approximately 510,000. assume the population is increasing at a rate of 4.6% per year.
a. write the exponential function that relates the total population, ( p(t) ), as a function of ( t ), the number of years since 2000.
( p(t)=510000(1.046)^{t} )
b. use part a. to determine the rate at which the population is increasing in ( t ) years. use exact expressions.
( p^{prime}(t)=510000(1.046)^{t} ln (1.046) ) people per year
c. use part b. to determine the rate at which the population is increasing in the year 2006. round to the nearest person per year.
( p^{prime}(6)=29835 ) people per year
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Explanation:

Step1: Substitute \(t = 6\) into \(P'(t)\)

We know that \(P'(t)=510000(1.046)^{t}\ln(1.046)\). For \(t = 6\), we have \(P'(6)=510000\times(1.046)^{6}\times\ln(1.046)\).
First, calculate \((1.046)^{6}\):
\((1.046)^{6}=e^{6\ln(1.046)}\approx e^{6\times0.0449}\) (since \(\ln(1.046)\approx0.0449\))
\(e^{6\times0.0449}=e^{0.2694}\approx1.309\)
Then, \(P'(6)=510000\times1.309\times0.0449\)
\(510000\times1.309 = 667590\)
\(667590\times0.0449\approx667590\times\frac{449}{10000}\)
\(667590\times449=(667590\times400)+(667590\times40)+(667590\times9)\)
\(667590\times400 = 267036000\)
\(667590\times40=26703600\)
\(667590\times9 = 6008310\)
\(267036000+26703600+6008310=300747910\)
\(300747910\div10000 = 30074.791\)

Answer:

\(P'(6)\approx30075\) people per year