Scientists have determined that when nutrients are sufficient, the number of bacteria grows exponentially. Suppose there are initially 1000 bacteria, and they increase to 3000 after ten minutes. How many bacteria are there after 1 hour?

Question

High School

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Answer :

Alenaa2 Alenaa2 · Answer 1 · 2025-03-05T20:00:19-05:00

After 1 hour, there are [tex]\(729000} \)[/tex] bacteria.

To determine how many bacteria there are after 1 hour, given the initial and growth conditions provided, we will use the exponential growth formula:

[tex]\[ N(t) = N_0 \cdot e^{kt} \][/tex]

where:

- [tex]\( N(t) \)[/tex] is the number of bacteria at time [tex]\( t \)[/tex]

- [tex]\( N_0 \)[/tex] is the initial number of bacteria,

- [tex]\( k \)[/tex] is the growth rate constant.

Given:

- Initial number of bacteria,[tex]\( N_0 = 1000 \).[/tex]

- Number of bacteria after 10 minutes (which is [tex]\( \frac{1}{6} \)[/tex] hours), [tex]( N\left(\frac{1}{6}\right) = 3000 \).[/tex]

Step 1: Find the growth rate constant [tex]\( k \).[/tex]

Using the given data at [tex]\( t = \frac{1}{6} \)[/tex] hours:

[tex]\[ N\left(\frac{1}{6}\right) = 1000 \cdot e^{k \cdot \frac{1}{6}} = 3000 \][/tex]

Divide both sides by 1000:

[tex]\[ e^{k \cdot \frac{1}{6}} = 3 \][/tex]

Take the natural logarithm of both sides to solve for [tex]\( k \):[/tex]

[tex]\[ k \cdot \frac{1}{6} = \ln(3) \][/tex]

Multiply both sides by 6:

[tex]\[ k = 6 \cdot \ln(3) \][/tex]

Step 2: Calculate the number of bacteria after 1 hour (which is 1 hour = 6 times 10 minutes).

Now use [tex]\( t = 1 \)[/tex] hour in the exponential growth formula:

[tex]\[ N(1) = 1000 \cdot e^{k \cdot 1} \][/tex]

[tex]\[ N(1) = 1000 \cdot e^{6 \cdot \ln(3)} \][/tex]

[tex]\[ N(1) = 1000 \cdot (e^{\ln(3)})^6 \][/tex]

[tex]\[ N(1) = 1000 \cdot 3^6 \][/tex]

Calculate[tex]\( 3^6 \):[/tex]

[tex]\[ 3^6 = 729 \][/tex]

Therefore,

[tex]\[ N(1) = 1000 \cdot 729 = 729000 \][/tex]