The number of tourists visiting a theme park increases every year and can be found using the equation 𝑦 is equal to 1.1 multiplied by 1.045 to the power of 𝑡, where 𝑦 million is the number of visitors 𝑡 years after 2010. If the number of visitors continues to increase at the same rate, in what year will the park first reach two million visitors?
The number of visitors can be found using the equation 1.1 multiplied by 1.045 to the power of 𝑡. 𝑦 is the number of visitors in millions and 𝑡 is the time after 2010. We need to work out the year when the number of visitors to the park reaches two million. For this to happen, 𝑦 must be greater than or equal to two. Therefore, 1.1 multiplied by 1.045 to the power of 𝑡 must be greater than or equal to two.
We could solve this problem by trial and improvement by substituting in values for 𝑡 until we get an answer greater than or equal to two. However, as this would be very time consuming, we will use logarithms to solve the inequality. Our first step is to divide both sides of the inequality by 1.1. The left-hand side becomes 1.045 to the power of 𝑡. The right-hand side becomes 1.8181 and so on. Next, we can take logs of both sides. Log of 1.45 to the power of 𝑡 is greater than or equal to log of 1.8181 and so on.
Next, we recall one of our laws of logarithms: log 𝑥 to the power of 𝑛 is equal to 𝑛 multiplied by log 𝑥. This means that we can rewrite the left-hand side as 𝑡 multiplied by log of 1.045. We will now clear some space to solve the rest of the problem. We can divide both sides of the inequality by log of 1.045. Typing the right-hand side into our calculator gives us 13.5819 and so on. As we’re looking for the number of years, we need to round this to the nearest whole number. 𝑡 must be greater than or equal to 13.5. This means that we must round up. We can, therefore, say that 𝑡 is equal to 14 years. This is the number of years after 2010 2010 plus 14 is equal to 2024.
We can, therefore, conclude that the year when the number of visitors reached two million is 2024.