A bacterial culture grows according to the formula $P = P _ { \mathrm { o } } a ^ { t }$. If it takes 4 days for the culture to triple in size, how long will it take to double in size?
A) $\quad - 0.9$ days
B) $\quad 2.1$ days
C) $\quad 0.4$ days
D) $\quad 5.0$ days
E) none of these

Question

Quizplus · Accepted Answer

We know that after 4 days, the population has tripled. Therefore:

$$3P_{o} = P_{o}a^{4}$$

Simplifying, we get:

$$a = 3^{1/4}$$

To find out how long it takes for the population to double, we need to find the value of t that satisfies the equation:

$$2P_{o} = P_{o}(3^{1/4})^{t}$$

Simplifying, we get:

$$2 = 3^{t/4}$$

Taking the ln of both sides yields:

$$\ln(2) = \frac{t}{4}\ln(3)$$

Solving for t, we get:

$$t = \frac{4\ln(2)}{\ln(3)} \approx 0.4$$

Therefore, it takes approximately 0.4 days for the culture to double in size.

A Bacterial Culture Grows According to the Formula P=PoatP = P _ { \mathrm { o } } a ^ { t }P=Po​at

A Bacterial Culture Grows According to the Formula $P = P _ { \mathrm { o } } a ^ { t }$