### 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.