Question

Growth of an Insect Population The size \(P\) of a certain insect population at time \(t\) (in days) obeys the law of uninhibited growth \(P(t)=500 e^{0.02 t}\) (a) Determine the number of insects at \(t=0\) days. (b) What is the growth rate of the insect population? (c) What is the population after 10 days? (d) When will the insect population reach \(800 ?\) (e) When will the insect population double?

Short answer

At t=0, the population is 500. The growth rate is 2% per day. After 10 days, the population is approximately 610.7. It will reach 800 in about 23.51 days. The population will double in about 34.65 days.

Step-by-step solution

- Determine Number of Insects at t=0 Days

To find the population at time zero, substitute t=0 into the given function: \[P(t) = 500 e^{0.02 t}\] So, \[P(0) = 500 e^{0.02 \times 0} = 500 e^{0} = 500\] Therefore, the number of insects at t=0 days is 500.

- Determine Growth Rate

The growth rate of the insect population is the exponent in the exponential function. In the equation \[P(t) = 500 e^{0.02 t}\] the growth rate is 0.02 or 2% per day.

- Determine Population After 10 Days

To calculate the population after 10 days, substitute t=10 into the function: \[P(t) = 500 e^{0.02 t}\] So, \[P(10) = 500 e^{0.02 \times 10} = 500 e^{0.2}\] Using a calculator, \[P(10) \text{ is approximately } 500 \times 1.2214 = 610.7\] Therefore, the population after 10 days is approximately 610.7 insects.

- Determine When the Population Will Reach 800

Set the given function equal to 800 and solve for t: \[800 = 500 e^{0.02 t}\] Divide both sides by 500: \[1.6 = e^{0.02 t}\] Take the natural logarithm of both sides: \[\text{ln}(1.6) = 0.02 t\] Solve for t: \[t = \frac{\text{ln}(1.6)}{0.02}\] Using a calculator, \[t \text{ is approximately } \frac{0.470}{0.02} = 23.51 \text{ days}\] Therefore, it will take approximately 23.51 days for the population to reach 800 insects.

- Determine When the Population Will Double

The initial population at t=0 is 500. To find when the population will double to 1000, set the function equal to 1000 and solve for t: \[1000 = 500 e^{0.02 t}\] Divide both sides by 500: \[2 = e^{0.02 t}\] Take the natural logarithm of both sides: \[\text{ln}(2) = 0.02 t\] Solve for t: \[t = \frac{\text{ln}(2)}{0.02}\] Using a calculator, \[t \text{ is approximately } \frac{0.693}{0.02} = 34.65 \text{ days}\] Therefore, it will take approximately 34.65 days for the population to double.

Key concepts

exponential functions

Exponential functions are mathematical expressions where a constant base is raised to a power that is a variable. These functions are used to model a wide range of phenomena, such as population growth, radioactive decay, and interest calculations. In its general form, an exponential function can be written as

• \(f(x) = a \times b^{x}\), where:
  • a is the initial value or starting point.
  • b is the base, which determines the rate of growth or decay.
  • x is the exponent, reflecting the variable of interest, often time.

In the provided exercise, the population of insects follows the exponential function \(P(t) = 500 e^{0.02 t}\), where:

• \(500\) is the initial population at \(t=0\) days.
• The growth rate, expressed in the exponent, is \(0.02\).

Exponential functions are distinct because they exhibit continuous and compounded growth or decay, leading to rapid changes over time.

growth rate

The growth rate in an exponential function represents the rate at which the quantity increases or decreases over time. Mathematically, it is the exponent in the function. For the insect population, the growth rate is \(0.02\) per day, or \(2\%\) per day. Growth rates can be positive (indicating an increase) or negative (indicating a decrease). The growth rate is crucial because it determines how quickly the population will grow:

• A higher growth rate leads to faster population growth.
• A lower growth rate results in slower population growth.

In the context of solving real-life problems such as the insect population exercise, identifying the growth rate helps predict future population sizes and assess how changing parameters might influence long-term trends.

natural logarithm

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e \approx 2.71828\). It is widely used in mathematics, particularly in dealing with exponential growth and decay problems. The core properties of the natural logarithm make solving equations involving exponents easier. For instance:

• Natural logarithms convert multiplication into addition: \(\ln(ab) = \ln(a) + \ln(b)\).
• They convert division into subtraction: \(\ln\left( \frac{a}{b} \right) = \ln(a) - \ln(b)\).
• They help isolate the exponent: If \(e^{x} = y\), then \(x = \ln(y)\).

In the insect population exercise, the natural logarithm helps determine when the population reaches a particular size or doubles. For example, to find when the population reaches \(800\) insects, we solve the equation \(500 e^{0.02t} = 800\) using \(\ln(1.6) = 0.02 t\) and solve for \(t\). This demonstrates the power and utility of natural logarithms in exponential growth problems.

population models

Population models are mathematical representations used to describe how populations change over time. They are vital in fields like biology, ecology, and environmental science. Exponential growth models, such as \(P(t) = P_0 e^{rt}\), describe populations growing at a constant relative rate. Here are some components of these models:

• Initial Population (\(P_0\)): The starting number of individuals.
• Growth Rate (\(r\)): The percentage increase per time unit.
• Time (\(t\)): The variable over which growth is measured.

For the insect population, \(P(t) = 500 e^{0.02 t}\) represents an exponential growth model where:

• \(500\) is the initial population.
• \(0.02\) is the growth rate or \(2\%\) per day.
• \(t\) is time measured in days.

Such models are powerful tools for predicting future population sizes and understanding the dynamics of population changes over time. They allow for proactive measures in conservation, resource management, and biodiversity studies.