Question

the number of bacteria in a culture is given by the function
\
(t) = 930e^{0.1t}\\
where \\(t\\) is measured in hours.
(a) what is the relative rate of growth of this bacterium population?
your answer is percent
(b) what is the initial population of the culture (at \\(t=0\\))?
your answer is
(c) how many bacteria will the culture contain at time \\(t=5\\)?
your answer is

Explanation:

🆕 New Concept Discovered: Exponential Growth Models
How populations grow continuously over time

Step 1: Identify the relative rate of growth

The exponential growth model is given by the formula:

$$ n(t) = n_0 e^{rt} $$

where:

• \( n_0 \) is the initial population.
• \( r \) is the relative rate of growth (expressed as a decimal).
• \( t \) is the time in hours.

Comparing this to our given function:

$$ n(t) = 930e^{0.1t} $$

The growth rate as a decimal is:

$$ r = 0.1 $$

To convert this decimal to a percentage, multiply by \( 100 \):

$$ 0.1 \times 100 = 10\% $$

Step 2: Find the initial population

The initial population occurs at time \( t = 0 \). We can find this by substituting \( t = 0 \) into the function:

$$ n(0) = 930e^{0.1(0)} $$

$$ n(0) = 930e^0 $$

Since any non-zero number raised to the power of \( 0 \) is \( 1 \) (\( e^0 = 1 \)):

$$ n(0) = 930 \times 1 = 930 $$

Step 3: Calculate the population at \( t = 5 \)

To find the population after \( 5 \) hours, substitute \( t = 5 \) into the function:

$$ n(5) = 930e^{0.1(5)} $$

$$ n(5) = 930e^{0.5} $$

Using a calculator to find the value of \( e^{0.5} \approx 1.64872 \):

$$ n(5) \approx 930 \times 1.64872 \approx 1533.31 $$

Depending on the rounding instructions of your online homework system, this is typically rounded to the nearest whole bacterium:

$$ n(5) \approx 1533 $$

Answer:

(a) 10 percent
(b) 930
(c) 1533 (or 1533.31 if decimals are required)