The Asian elephant population 𝑡 years after the year 1900 is given by 𝑃 equals 100,000 times 0.25 to the power of 𝑡 over 100. What was the Asian elephant population in 1900? According to this model, by what percentage has the Asian elephant population decreased over a century?
Let’s begin by inspecting the equation we’ve been given. 𝑃 equals 100,000 times 0.25 to the power of 𝑡 over 100. This is an exponential equation. Remember, an exponential equation or function is of the form 𝑎 times 𝑏 to the 𝑥 power. We also know that 𝑏 must be greater than zero and not equal to one. The exact size of 𝑏 tells us whether we’re working with exponential growth or decay. Now, if we compare our general form with the equation for the population, we see that 𝑏 is equal to 0.25. Since 𝑏 is between zero and one, we must have exponential decay. That is, the Asian elephant population decreases year on year.
Now we want to find the size of the Asian elephant population in 1900. Well, the equation gives us the population 𝑡 years after the year 1900. It follows then that 1900 is actually zero years after 1900. So one method we have to find the population in 1900 is to substitute 𝑡 equals zero into our equation. We get 100,000 times 0.25 to the power of zero over 100. Well, zero divided by 100 is zero, and then some number to the power of zero is equal to one. So the population is 100,000 times one, which is simply 100,000. The elephant population in 1900 then was 100,000.
Now we could have determined this directly from the equation. When we have an equation in the form 𝑎 times 𝑏 to the 𝑥 power, the value of 𝑎 is actually the initial value. It tells us the value when 𝑥 is equal to zero. So in this case, we could have determined this by simply finding the value of 𝑎, which is 100,000. The next part of this question asks us to find the percentage by which the Asian elephant population decreases over a century. Well, of course, a century is equal to 100 years. So we can substitute 𝑡 equals 100 into this equation. When we do, we find that 𝑃 is equal to 100,000 times 0.25 to the power of 100 divided by 100. But 100 divided by 100 is simply equal to one. So the population must be 100,000 times 0.25.
Now, we could work this out and then work out this number as a percentage of the original. But actually a little bit of knowledge about how exponential decay works will allow us to answer this question without doing so. Each century we see that the original population, 100,000, is multiplied by 0.25. Multiplying 0.25 by 100 and we get 25 percent. This means a century after the year 1900, the population is 25 percent of the original population.
But of course, finding 25 percent of the original number means that we must have reduced that original number by 75 percent. 100 percent minus 75 percent is 25 percent. So for the population to be 25 percent of the size it was in 1900 means it must have reduced by 75 percent. And so we’ve answered the question. In 1900, the population was 100,000 elephants. And according to this model, the percentage by which the population has decreased over a century is 75 percent.