# Video: Pack 1 • Paper 3 • Question 21

Pack 1 • Paper 3 • Question 21

02:05

### Video Transcript

A new antibiotic is being tested on a culture of bacteria. Initial results suggest that 𝑡 hours after the antibiotic is administered, the number of bacteria is 10 to the power of nine multiplied by three to the power of negative 𝑡. Part a) Approximately, how many bacteria will there be five hours after the antibiotic is administered? Part b) Will the number of bacteria ever reach zero? Explain your answer.

In order to answer the first part of the question, we need to substitute 𝑡 equals five into the equation 10 to the power of nine multiplied by three to the power of negative 𝑡. Substituting in 𝑡 equals five give us 10 to the power of nine multiplied by three to the power of negative five. Typing this into our calculator gives us 4115226.337. Rounding this to the nearest whole number means that the number of bacteria after five hours is approximately 4115226.

The second part of our question asked whether the number of bacteria will ever reach zero. Well, despite the fact that the number of bacteria are decreasing, we can never get to zero when dividing 10 to the power of nine by a positive number. Three to the power of negative 𝑡 is the same as one over three to the power of 𝑡. 10 to the power of nine divided by three to the power of 𝑡 can never equal zero. Therefore, the number of bacteria will never reach zero. Dividing a positive number by a positive number will always give a positive answer. In this case, no matter what the value of 𝑡, we’ll never reach zero.