The population of a rare orchid declines by 𝑟 percent every year. There are currently only 99 of these orchids left, and conservationists predict that only 50 will be left in five years. Write an equation that can be used to find 𝑟, the rate of decline.
In this question, there are 99 orchids, which is a type of flower. Every year, the number of orchids decreases by 𝑟 percent. After five years, we’re told that only 50 orchids remain. We need to write an equation to find 𝑟, the rate of decline. It’s important to note that because 𝑟 is a percentage, then the decrease won’t be a fixed value each year. We can solve this problem by using the exponential function most commonly seen to answer problems involving the compound interest in a banking situation. This formula tells us that 𝑉, the value after 𝑦 years, is equal to 𝑃, which is the starting amount, multiplied by one plus 𝑟, the rate over 100, to the power of 𝑦, which is the number of years.
Before we rush in to filling in the values into this formula, there’s one important change we need to make to this formula. Because the population of orchids is not increasing, this plus needs to be a subtraction because we’re decreasing by a certain rate each year. Filling the values into the formula then, the value at the end of five years is 50. The starting amount is 99 orchids. We’re told to use 𝑟 as the rate of decline, and the number of years we’re given is five years. As we’re asked to give an equation that could be used to find 𝑟, then we could give the answer 99 times one minus 𝑟 over 100 to the power of five equals 50.
This equation or the equation above would both be perfectly valid answers. We weren’t actually asked to calculate 𝑟, the rate of decline, but we could do so by rearranging this equation. Doing so would enable us to work out that 𝑟, the rate of decline, is approximately 13 percent each year. However, all we needed to do is to give the equation that could be used to find 𝑟.