A bacteria culture starts with 
 bacteria and grows at a rate proportional to its size. After 
 hours there will be 
 bacteria.

(a) Express the population 
 after 
 hours as a function of 
. Be sure to keep at least 4 significant figures on the growth rate.
= 
  

(b) What will be the population after 8 hours?
   = how much bacteria

(c) How long will it take for the population to reach 1950? Give your answer accurate to at least 2 decimal places.
  how many hours

Let's break this problem down and solve it step by step.

We know that the bacteria culture grows at a rate proportional to its size, which indicates exponential growth. The general form of the population growth equation is:

\[
P(t) = P_0 e^{kt}
\]

Where:
- \( P(t) \) is the population after \( t \) hours,
- \( P_0 \) is the initial population,
- \( k \) is the growth rate constant, and
- \( t \) is the time in hours.

### Part (a) - Express the population \( P(t) \) after \( t \) hours as a function of \( t \)

To find the value of \( k \), we need to use the information that after a certain number of hours \( t \), the population reaches a certain value.

Given dataLet's break this problem down and solve it step by step.

We know that the bacteria culture grows at a rate proportional to its size, which indicates exponential growth. The general form of the population growth equation is:

\[
P(t) = P_0 e^{kt}
\]

Where:
- \( P(t) \) is the population after \( t \) hours,
- \( P_0 \) is the initial population,
- \( k \) is the growth rate constant, and
- \( t \) is the time in hours.

### Part (a) - Express the population \( P(t) \) after \( t \) hours as a function of \( t \)

To find the value of \( k \), we need to use the information that after a certain number of hours \( t \), the population reaches a certain value.

Given 
- Initial population \( P_0 \),
- Population after \( t \) hours.

We can substitute these values into the formula and solve for \( k \).

### Part (b) - Population after 8 hours

Once we know \( k \), we can use the equation \( P(t) = P_0 e^{kt} \) to find the population after 8 hours by substituting \( t = 8 \).

### Part (c) - Time to reach a population of 1950

To solve for \( t \) when \( P(t) = 1950 \), we'll rearrange the exponential equation and solve for \( t \). This requires taking the natural logarithm on both sides of the equation.

Let me fill in the data based on your placeholders and calculate each part. Could you provide the missing numerical values (initial population, population after a certain time, and the time)?

A bacteria culture starts with an initial population and grows at a rate proportional to its size. After a given number of hours, the population reaches a certain value. (a) Express the population after t hours as a function of t. (b) What will be the population after 8 hours? (c) How long will it take for the population to reach 1950?

Learn how to model bacterial growth using exponential functions. This problem covers key concepts in exponential growth and natural logarithms, demonstrating how to calculate the growth rate, predict population after 8 hours, and find the time needed for a population to reach 1950. Suitable for students in Grades 10-12.

Math Problem Statement

(a) Express the population after hours as a function of . Be sure to keep at least 4 significant figures on the growth rate.

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