A population of bacteria is growing according to the equation , where is
the number of hours.
Estimate when the population will exceed 648 .

Give your answer accurate to one decimal place.
Check Answer

Ask by Phillips Reeves. in the United States

Nov 24,2024

Question

A population of bacteria is growing according to the equation , where is
the number of hours.
Estimate when the population will exceed 648 .

Give your answer accurate to one decimal place.
Check Answer

Ask by Phillips Reeves. in the United States

Nov 24,2024

UpStudy AI · Accepted Answer

To find when the population $$ P(t) $$ exceeds 648, we need to solve the inequality:

$$
300 e^{0.22 t} > 648
$$

First, we can divide both sides by 300:

$$
e^{0.22 t} > \frac{648}{300}
$$

Calculating the right side:

$$
\frac{648}{300} = 2.16
$$

Now we have:

$$
e^{0.22 t} > 2.16
$$

Next, we take the natural logarithm of both sides:

$$
0.22 t > \ln(2.16)
$$

Calculating $$ \ln(2.16) $$:

$$
\ln(2.16) \approx 0.775
$$

Now we can solve for $$ t $$:

$$
0.22 t > 0.775
$$

Dividing both sides by 0.22:

$$
t > \frac{0.775}{0.22} \approx 3.523
$$

Rounding to one decimal place, we find:

$$
t \approx 3.5
$$

Thus, the population will exceed 648 at approximately $$ t = 3.5 $$ hours.
The population will exceed 648 at approximately $$ t = 3.5 $$ hours.