https://file.aiquickdraw.com/mathbot/file/1726297279023hotjqfue.jpg

The exercise you uploaded includes questions related to population growth using various rates and timeframes, as well as a question about bacteria growth. Let's break down and solve each of the problems in the exercise:

### Formula for population growth:
The general formula for population growth (exponential growth) is:

\[
A = P \times (1 + \frac{r}{100})^t
\]

Where:
- \( A \) = Amount after growth
- \( P \) = Initial population (Principal)
- \( r \) = Growth rate per year (in %)
- \( t \) = Time (in years)

#### 1. P = 680,000; r = 12% per year; t = 8 years
\[
A = 680,000 \times (1 + \frac{12}{100})^8
\]

#### 2. A = 1,240,000; r = 8% per year; t = 30 years
To find \( P \) (initial population), we rearrange the formula:
\[
P = \frac{A}{(1 + \frac{r}{100})^t}
\]

\[
P = \frac{1,240,000}{(1 + \frac{8}{100})^{30}}
\]

#### 3. A = 786,000; P = 247,000; t = 17 years
To find the growth rate \( r \), we rearrange the formula:
\[
r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1
\]

\[
r = \left(\frac{786,000}{247,000}\right)^{\frac{1}{17}} - 1
\]

#### 4. A = 731,093; P = 525,600; r = 3% per year
To find the time \( t \), we rearrange the formula:
\[
t = \frac{\log(\frac{A}{P})}{\log(1 + \frac{r}{100})}
\]

\[
t = \frac{\log(\frac{731,093}{525,600})}{\log(1 + \frac{3}{100})}
\]

#### 5. Suppose the population of bacteria doubles every 6 hours. If the initial population is 100, and we want to know how many bacteria will be there in two days:
- Doubling every 6 hours means every 6 hours the population is multiplied by 2.
- In 2 days (48 hours), there will be \( \frac{48}{6} = 8 \) periods of doubling.

The population after 2 days will be:
\[
A = 100 \times 2^8 = 100 \times 256 = 25,600
\]

Let me now calculate the results for you.Here are the solutions for each problem:

1. **Final population after 8 years** (P = 680,000; r = 12%; t = 8 years):
   \[
   A = 1,683,655 \, \text{(rounded to nearest whole number)}
   \]

2. **Initial population** for an ending population of 1,240,000 (A = 1,240,000; r = 8%; t = 30 years):
   \[
   P = 123,228 \, \text{(rounded to nearest whole number)}
   \]

3. **Growth rate** required for population growth from 247,000 to 786,000 over 17 years:
   \[
   r \approx 7.05\% \, \text{per year}
   \]

4. **Time required** for the population to grow from 525,600 to 731,093 at a 3% annual growth rate:
   \[
   t \approx 11.16 \, \text{years}
   \]

5. **Bacteria population after two days** (starting with 100 and doubling every 6 hours):
   \[
   \text{Final population} = 25,600 \, \text{bacteria}
   \]

Would you like further explanations or calculations? Here are 5 additional questions to expand your understanding:

1. How would changing the growth rate affect the population after a set number of years?
2. What happens to the doubling period in problem 5 if the bacteria doubles every 3 hours instead of 6?
3. How is exponential growth different from linear growth?
4. What formula adjustments are needed for continuous (as opposed to discrete) growth?
5. How can we account for decay instead of growth in population models?

**Tip**: Exponential growth models are sensitive to changes in rate. Small changes in growth rate can result in large differences over time.

Solve and substitute the given values in the formula of population growth:
1. P=680,000; r=12% per year; t=8 years
2. A=1,240,000; r=8% per year; t=30 years
3. A=786,000; P=247,000; t=17 years
4. A=731,093; P=525,600; r=3% per year
5. Suppose the population of a certain bacteria in a laboratory sample is 100. If it doubles in population every 6 hours, what is the growth rate? How many bacteria will be there in two days?

Learn how to solve problems related to population growth and doubling time using exponential growth formulas. Includes examples of bacterial population doubling and population growth over time. Suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation