A study of 10000 people was carried out to investigate the rate of infection of influenza. The number of infections, 𝑦, occurring 𝑛 years after 2004 can be found using the equation 𝑦 = −2.5𝑛² − 7.5𝑛 + 909. Calculate the number of infections in 2010 and 2012.

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### Video Transcript

A study of 10000 people was carried out to investigate the rate of infection of influenza. The number of infections 𝑦 occurring 𝑛 years after 2004 can be found using the equation 𝑦 equals negative 2.5𝑛 squared minus 7.5𝑛 plus 909. Calculate the number of infections in 2010 and 2012.

Let’s start this question by looking at our equation. We’re told that 𝑦 is the number of infections of influenza and 𝑛 is the number of years after 2004. We’re asked to calculate the number of infections in 2010 and 2012. That means we’re going to calculate 𝑦 for the 𝑛-value related to 2010 and the 𝑛-value related to 2012.

So let’s start by looking at 2010. It might be tempting when we look at the year 2010 to use an 𝑛-value of 2010. We must be careful and notice that the 𝑛-value is the number of years after 2004. So we must work out how many years after 2004 that the year 2010 is. And that’s six. And now we know that 𝑛 must be equal to six.

Taking our equation then and substituting the value of 𝑛 equals six will give us 𝑦 equals negative 2.5 times six squared minus 7.5 times six plus 909. Simplifying our terms then, since we have six squared equal to 36, and minus 2.5 times 36 is negative 19. We can calculate negative 7.5 times six by working out seven times six, which is 42, and then adding on another half of six, which is three, giving us minus 45 and plus 909.

And therefore, we have 𝑦 equals 774. We know that our 𝑦-value is the number of infections in the study of 10000 people. So we can say, in 2010, the number of infections is 774 persons.

Moving on then to look at the year 2012 or 2012, to work on our 𝑛-value then, we establish that 2012 is eight years after 2004. So our 𝑛 value is eight. Substituting 𝑛 equals eight into our equation gives us 𝑦 equals negative 2.5 times eight squared minus 7.5 times eight plus 909. Simplifying our right-hand side, we have negative 2.5 times 64, which is negative 160, subtract 60 plus 909. Therefore, 𝑦 equals 689 persons.

So our final answer for the number of infections in 2010 and 2012 is 774 persons and 689 persons.