Question

a graphing calculator is recommended.
a bacterial culture starts with 700 bacteria and doubles in size every half - hour.
(a) how many bacteria are there after 2 hours?
(b) how many bacteria are there after t hours?
(c) how many bacteria are there after 40 minutes? (round your answer to the nearest whole number)
(d) graph the population function.

Explanation:

Step1: Determine the number of half - hour intervals

Since the bacteria doubles every half - hour, in 2 hours, there are $\frac{2}{0.5}=4$ half - hour intervals.

Step2: Use the exponential growth formula

The initial number of bacteria $a = 700$, and the growth factor $b = 2$. The formula for the number of bacteria $y$ after $n$ half - hour intervals is $y=a\times b^{n}$. Substituting $a = 700$, $b = 2$ and $n = 4$ into the formula, we get $y=700\times2^{4}=700\times16 = 11200$.

Step3: Find the formula for $t$ hours

In $t$ hours, the number of half - hour intervals $n = 2t$. So the formula for the number of bacteria $y$ after $t$ hours is $y = 700\times2^{2t}=700\times4^{t}$.

Step4: Determine the number of half - hour intervals in 40 minutes

40 minutes $=\frac{40}{30}=\frac{4}{3}$ half - hour intervals. Using the formula $y = 700\times2^{n}$, substituting $n=\frac{4}{3}$, we have $y=700\times2^{\frac{4}{3}}=700\times\sqrt[3]{16}\approx700\times2.52=1764$.

Answer:

(a) 11200 bacteria
(b) $y = 700\times4^{t}$ bacteria
(c) 1764 bacteria
(d) To graph $y = 700\times4^{t}$, the $y$ - intercept is at $(0,700)$. As $t$ increases, the function is an exponential growth function. The graph starts at the point $(0,700)$ and rises rapidly as $t$ moves to the right.