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The number of tourists visiting a theme park increases every year and can be found using the equation π¦ is equal to 1.1 multiplied by 1.045 to the power of π‘, where π¦ million is the number of visitors π‘ years after 2010.
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If the number of visitors continues to increase at the same rate, in what year will the park first reach two million visitors?
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The number of visitors can be found using the equation 1.1 multiplied by 1.045 to the power of π‘.
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π¦ is the number of visitors in millions and π‘ is the time after 2010.
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We need to work out the year when the number of visitors to the park reaches two million.
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For this to happen, π¦ must be greater than or equal to two.
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Therefore, 1.1 multiplied by 1.045 to the power of π‘ must be greater than or equal to two.
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We could solve this problem by trial and improvement by substituting in values for π‘ until we get an answer greater than or equal to two.
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However, as this would be very time consuming, we will use logarithms to solve the inequality.
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Our first step is to divide both sides of the inequality by 1.1.
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The left-hand side becomes 1.045 to the power of π‘.
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The right-hand side becomes 1.8181 and so on.
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Next, we can take logs of both sides.
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Log of 1.45 to the power of π‘ is greater than or equal to log of 1.8181 and so on.
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Next, we recall one of our laws of logarithms: log π₯ to the power of π is equal to π multiplied by log π₯.
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This means that we can rewrite the left-hand side as π‘ multiplied by log of 1.045.
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We will now clear some space to solve the rest of the problem.
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We can divide both sides of the inequality by log of 1.045.
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Typing the right-hand side into our calculator gives us 13.5819 and so on.
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As weβre looking for the number of years, we need to round this to the nearest whole number.
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π‘ must be greater than or equal to 13.5.
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This means that we must round up.
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We can, therefore, say that π‘ is equal to 14 years.
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This is the number of years after 2010 2010 plus 14 is equal to 2024.
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We can, therefore, conclude that the year when the number of visitors reached two million is 2024.