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6.7 problem 7
Suppose P=f(t) is the population (in thousands) of town t years after 1990, and that f(7)=15 and f(12)=25,

(a) Find a formula for f(t) assuming f is exponential in the form ab^t. Use 5 decimal places for a & b: P=f(t)=

(b) Find a formula for f−1(P)=

(c) Evaluate f(45)=

(d) f−1(45)=

Write out sentences to explain the practical meaning of your answers to parts (c) and (d). Consider the seven numbered statements in the list below:

The town's population in 2045 is f(45) people.
The town's population has grown by f(45) people over a 45 year period.
The town's population in 2035 is f(45) people.
The town's population will reach 45,000 people in f−1(45) years after 1990.
The town's population will reach 45,000 people in f−1(45) years from now.
The town's population will reach 45 people in f−1(45) years after 1990.
The town's population in 2035 is f(45) thousand people.

(e) Which statement above explains the meaning of your answer to (c)? (enter the number 1-7 of the correct statement).

(f) Which statement above explains the meaning of your answer to (d)? (enter the number 1-7 of the correct statement).

Respuesta :

lhmarianateixeira

So the corresponding answers to these questions are:

[tex]A) a= 7.33657 \\ b= 1.10757\\B)f(t)^{-1} = log (7.33657)+ t*log(1.10757)\\C)728.10886\\D) 2.86219[/tex]

For the letter a, it will be necessary to calculate the values ​​of a and b, therefore:

[tex]f(t)=a*b^{t} \\f(7)=15 \\ a*b^{7}=15\\a=\frac{15}{b^{7}} \\f(12)=25\\a= \frac{25}{b^{12}}\\\frac{15}{b^{7}} =\frac{25}{b^{12}}=> b^{5}= 1.66667=> b= 1.10757\\a=\frac{15}{b^{7}}=> a=7.33657[/tex]

So for the letter B we will do the logarithm so we will have:

[tex]P=f(t)=ab^{t}=(7.33657)(1.10757)^{t} \\f(t)^{-1} = log (7.33657)+ t*log(1.10757)[/tex]

For the letter C we will use the formula given in the statement just substituting the value of t=45:

[tex]f(t)=ab^{t} \\f(45)=(7.33657)(1.10757)^{45} = 728.10886[/tex]

The formula calculated on the letter B will be useful for the letter D only having to substitute t=45, therefore:

[tex]f(45)^{-1} = log (7.33657)+ 45*log(1.10757)\\f(45)^{-1} = 2.86219[/tex]

Learn more: brainly.com/question/20838017

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