Every ten years, the Bureau of the Census counts the number of people living in the United States. In 1790, the population of the U.S. was 3.93 million. By 1800, this number had grown to 5.31 million.

Write an exponential function that could be used to model the U.S. population y in millions for 1790 to 1800. Write the equation in terms of x, the number of decades since 1790.

Question

Every ten years, the Bureau of the Census counts the number of people living in the United States.  In 1790, the population of the U.S.  was 3.93 million.  By 1800, this number had grown to 5.31 million.  

Write an exponential function that could be used to model the U.S. population y in millions for 1790 to 1800.  Write the equation in terms of x, the number of decades since 1790.

ankit042 · Answer

y = ke^x +c 
now we know c = 3.93
y=5.31 for x =10 find value of k

ybarrap · Answer

$$y(t)=Kexp ^{\lambda t}$$
$$y(0)=K=3.93$$
$$y(1)=3.93\exp ^{\lambda (1)}=5.31$$
From this you can get $\lambda$ and you're done.

anonymous · Answer

Do I just solve normally?

ankit042 · Answer

sorry I think @ybarrap method is correct. I made a mistake

anonymous · Answer

Help? What do I do with the last part?

ybarrap · Answer

yes, solve for the missing variable, $ \lambda $. Do you know how?

anonymous · Answer

No... :(

anonymous · Answer

I don't really know what that little wish bone signifies.

ybarrap · Answer

$$\exp ^{\lambda  (1)}=\frac{ 5.31 }{ 3.93 }$$
$$\lambda=\ln(\frac{ 5.31 }{ 3.93 })$$

ybarrap · Answer

$ \lambda $ is the average growth rate over 10 years.

ybarrap · Answer

In millions of people, per decade.

anonymous · Answer

oooohhh I see

mathstudent55 · Answer

@ybarrap Isn't the data point Population = 5,310,000 at year 1800 a data point at x = 10? Why do you have $ \lambda(1) $?

ybarrap · Answer

Because the question asked about " Write the equation in terms of x, the number of decades since 1790." So the "1" is the 1st decade since 1790.

mathstudent55 · Answer

Good point. Thanks.

ybarrap · Answer

np