### Video Transcript

The US census is taken every 10
years. The population of Texas was 3.05
million in 1900 and 20.9 million in 2000. By modeling the growth as
exponential, answer the following questions. Write an exponential function in
the form π of π equals π nought times π to the π power to model the population
of Texas, in millions, π decades after 1900. Round your value of π to three
decimal places if necessary. According to the model, what was
the population of Texas in 1950? Give your answer to three
significant figures. And finally, rewrite your function
in the form π of π¦ is equal to π nought times π to the π¦ power, where π¦ is the
time in years after 1900. Round your value of π to four
decimal places.

Letβs start with what we know. In 1900, the population was 3.05
million. And by 2000, that value was 20.9
million. We know the general form of the
exponential function π of π₯ equals π΄ times π to the π₯ power. Weβre following this general form
with the function π of π equals π nought times π to the π power, where our π
represents time in decades after 1900. And that means our initial value π
nought should be the population in 1900. Since weβre working in millions, we
can leave this as 3.05. π is our unknown value. To solve for π, we can use our
other data point.

We know that in 2000 there was a
population of 20.9 million. We also know that 2000 is 10
decades after 1900. Plugging all of this in, we can use
that information to solve for π. To get π by itself, we divide both
sides of the equation by 3.05, which gives us 6.8524 continuing is equal to π to
the 10th power. Instead of rounding this, weβll
just leave it in our calculator as is. To isolate π, weβll raise both
sides of this equation to the one-tenth power. π to the 10th power to the
one-tenth power equals π, and 6.8524 continuing to the one-tenth power equals
1.212228 continuing. We want to round this π-value to
three decimal places.

The fourth decimal place has a
two. So we say that π equals 1.212. This π-value thatβs greater than
one tells us weβre dealing with population growth. And if we think of the decimal
0.212 as a percent, we can say that the population is growing at a rate of about
21.2 percent every decade. And weβve created a model to
calculate what the population would be π decades after 1900. π of π equals 3.05 times 1.212 to
the π power. Using this model, we want to
estimate what the population was in 1950. 1950 is 50 years after 1900, which
is five decades. To calculate this, we wanna take π
of five, 3.05 times 1.212 to the fifth power, which is equal to 7.9765
continuing. Three significant figures in this
case would be to the second decimal place. If we round to the second decimal
place, we get 7.98. Based on our model, we can expect
that the population of Texas in 1950 would have been 7.98 million.

For part three of this question, we
want to rewrite our exponential model, where our unit of time is years instead of
decades. It will be a really similar process
to what we did in the first part. Weβll still have the same initial
value of 3.05. But to solve for π¦, weβll use our
second data point. In 2000, the population was 20.9
million. And that was 100 years after our
initial value. So weβll plug in 100 for π¦, and
then we can solve for π. By dividing both sides of the
equation by 3.05, we get 6.8524 continuing equals π to the one hundredth power. Again, we donβt wanna round this
6.852 continuing yet. Weβll just leave it in our
calculator so that we can raise both sides of this equation to the one over 100
power.

π to the one hundredth power to
the one over 100 power equals π. And 6.8524 continuing to the one
over 100 power equals 1.01943 continuing. Weβre rounding to four decimal
places this time, which means weβll use π equals 1.0194. Our π-value is less than our
π-value. According to the π-value, the
population growth per year was 1.94 percent versus a 21.2 percent population growth
decade over decade. So for our yearly model, we have π
of π¦ equals 3.05 times 1.0194 to the π¦ power.