The size of a certain insect population is given by ​P(t)equals300e Superscript . 01 t​, where t is measured in days. At what time will the population equal 1500​?
Question content area bottom
Part 1
It will take 
  
enter your response here days for the population to equal 1500. 
​(Round to one decimal place as​ needed.)

We are given the population equation \( P(t) = 300e^{0.01t} \), where \( P(t) \) is the population at time \( t \) in days, and we want to find the time \( t \) when the population reaches 1500.

We set \( P(t) = 1500 \) and solve for \( t \).

\[
1500 = 300e^{0.01t}
\]

### Step 1: Divide both sides by 300
\[
\frac{1500}{300} = e^{0.01t}
\]
\[
5 = e^{0.01t}
\]

### Step 2: Take the natural logarithm (ln) of both sides
\[
\ln(5) = \ln(e^{0.01t})
\]
\[
\ln(5) = 0.01t
\]

### Step 3: Solve for \( t \)
\[
t = \frac{\ln(5)}{0.01}
\]

Now, we can compute this:

\[
t = \frac{\ln(5)}{0.01} \approx \frac{1.6094}{0.01} = 160.94
\]

Rounding to one decimal place, \( t \approx 160.9 \) days.

Thus, it will take approximately **160.9 days** for the population to reach 1500.

Would you like more details or further clarification?

Here are 5 questions that expand upon this concept:
1. How does the population change over time when \( t \) increases?
2. What would the population be at \( t = 0 \) days?
3. How would you find the rate of growth of the population at any given time \( t \)?
4. If the growth rate constant changed to 0.02, how would that affect the time to reach 1500?
5. How long would it take for the population to double?

**Tip:** Always check the base of the logarithm you use to ensure it's appropriate for the problem (e.g., natural logarithm for base \( e \)).

The size of a certain insect population is given by ​P(t) = 300e^0.01t​, where t is measured in days. At what time will the population equal 1500?

Find out how long it will take for an insect population to grow from 300 to 1500 using the exponential growth formula P(t) = 300e^0.01t. This solution involves key mathematical concepts such as logarithms and exponential functions, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation