Question Video: Creating Exponential Equations and Using Them to Solve Problems Mathematics

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 𝑃 of 𝑑 equals 𝑃 nought times 𝑘 to the 𝑑 power to model the population of Texas, in millions, 𝑑 decades after 1900. Round your value of 𝑘 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 𝑃 of 𝑦 is equal to 𝑃 nought times 𝑏 to the 𝑦 power, where 𝑦 is the time in years after 1900. Round your value of 𝑏 to four decimal places.

Let’s 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 𝑓 of 𝑥 equals 𝐴 times 𝑏 to the 𝑥 power. We’re following this general form with the function 𝑃 of 𝑑 equals 𝑃 nought times 𝑘 to the 𝑑 power, where our 𝑑 represents time in decades after 1900. And that means our initial value 𝑃 nought should be the population in 1900. Since we’re working in millions, we can leave this as 3.05. 𝑘 is our unknown value. To solve for 𝑘, 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 𝑘. To get 𝑘 by itself, we divide both sides of the equation by 3.05, which gives us 6.8524 continuing is equal to 𝑘 to the 10th power. Instead of rounding this, we’ll just leave it in our calculator as is. To isolate 𝑘, we’ll raise both sides of this equation to the one-tenth power. 𝑘 to the 10th power to the one-tenth power equals 𝑘, and 6.8524 continuing to the one-tenth power equals 1.212228 continuing. We want to round this 𝑘-value to three decimal places.

The fourth decimal place has a two. So we say that 𝑘 equals 1.212. This 𝑘-value that’s greater than one tells us we’re 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 we’ve created a model to calculate what the population would be 𝑑 decades after 1900. 𝑃 of 𝑑 equals 3.05 times 1.212 to the 𝑑 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 𝑃 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. We’ll still have the same initial value of 3.05. But to solve for 𝑦, we’ll use our second data point. In 2000, the population was 20.9 million. And that was 100 years after our initial value. So we’ll plug in 100 for 𝑦, and then we can solve for 𝑏. By dividing both sides of the equation by 3.05, we get 6.8524 continuing equals 𝑏 to the one hundredth power. Again, we don’t wanna round this 6.852 continuing yet. We’ll just leave it in our calculator so that we can raise both sides of this equation to the one over 100 power.

𝑏 to the one hundredth power to the one over 100 power equals 𝑏. And 6.8524 continuing to the one over 100 power equals 1.01943 continuing. We’re rounding to four decimal places this time, which means we’ll use 𝑏 equals 1.0194. Our 𝑏-value is less than our 𝑘-value. According to the 𝑏-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 𝑃 of 𝑦 equals 3.05 times 1.0194 to the 𝑦 power.