Question

shelly is a biologist who wants to determine the growth rate of a strain of bacteria. shelly puts 85 bacteria into a controlled environment and waits 4 hours. when shelly checks 4 hours later, there are 850 bacteria. what is the hourly growth rate of the bacteria? enter your answer as a number, rounded to hundredths, like this: 425.36

Explanation:

Step1: Use the exponential - growth formula

The general formula for exponential growth is $N = N_0\cdot(1 + r)^t$, where $N$ is the final amount, $N_0$ is the initial amount, $r$ is the growth rate per time - period, and $t$ is the number of time - periods. Here, $N_0 = 85$, $N = 850$, and $t = 4$. So, $850=85\cdot(1 + r)^4$.

Step2: Simplify the equation

Divide both sides of the equation $850 = 85\cdot(1 + r)^4$ by 85. We get $\frac{850}{85}=(1 + r)^4$, which simplifies to $10=(1 + r)^4$.

Step3: Solve for $(1 + r)$

Take the fourth - root of both sides. $1 + r=\sqrt[4]{10}$. Using a calculator, $\sqrt[4]{10}\approx1.7783$.

Step4: Solve for $r$

Subtract 1 from both sides to find $r$. $r=\sqrt[4]{10}-1\approx1.7783 - 1=0.7783$.

Step5: Round the answer

Rounding $0.7783$ to the hundredths place, we get $0.78$.

Answer:

$0.78$