Question

a scientist initially measures 12,000 bacteria in a growth medium. 4 hours later, the scientist measures 24,000 bacteria. assuming exponential growth, the formula $p = c(2)^{rt}$ gives the number of bacteria in the growth medium, where $r$ and $c$ are constants and $p$ is the number of bacteria $t$ hours after the initial measurement. what is the value of $r$?
(a) $\frac{1}{12000}$
(b) $\frac{1}{4}$
(c) 4
(d) 12,000

Explanation:

Step1: Find the initial - value

When \(t = 0\), \(P(0)=C(2)^{r\times0}=C\). Since initially (\(t = 0\)) there are 12000 bacteria, so \(C = 12000\). The formula becomes \(P = 12000(2)^{rt}\).

Step2: Substitute the values of \(t\) and \(P\)

We know that when \(t = 4\), \(P=24000\). Substitute these values into the formula \(P = 12000(2)^{rt}\), we get \(24000=12000(2)^{4r}\).

Step3: Simplify the equation

Divide both sides of the equation \(24000 = 12000(2)^{4r}\) by 12000. We have \(\frac{24000}{12000}=(2)^{4r}\), which simplifies to \(2=(2)^{4r}\).

Step4: Solve for \(r\)

Since the bases are the same (\(a^m=a^n\) implies \(m = n\) for \(a>0,a
eq1\)), when \(2=(2)^{4r}\), we can set the exponents equal: \(4r = 1\). Then solve for \(r\), we get \(r=\frac{1}{4}\).

Answer:

B. \(\frac{1}{4}\)