For The Function $f(t) = P E^{r T}$, If $P = 3$ And $r = 0.03$, Then What Is The Value Of $f(3$\] To The Nearest Tenth?A. 2.5 B. 3.3 C. 1.1 D. 7.4

Understanding the Function and Calculating f(3)

Introduction

In this article, we will explore the function $f(t) = P e^{r t}$ and calculate the value of $f(3)$ given that $P = 3$ and $r = 0.03$. This function is a type of exponential function, which is commonly used in finance, physics, and other fields to model growth and decay.

The Exponential Function

The exponential function is defined as $f(t) = P e^{r t}$, where $P$ is the initial value, $r$ is the growth rate, and $t$ is the time. In this case, we are given that $P = 3$ and $r = 0.03$. We need to calculate the value of $f(3)$, which means we need to find the value of the function when $t = 3$.

Calculating f(3)

To calculate $f(3)$, we need to substitute $t = 3$ into the function $f(t) = P e^{r t}$. This gives us:

$f(3) = 3 e^{0.03 \times 3}$

To evaluate this expression, we need to calculate the value of $e^{0.03 \times 3}$. This can be done using a calculator or by using the properties of exponents.

Evaluating the Exponential Expression

Using a calculator, we can evaluate the expression $e^{0.03 \times 3}$ as follows:

$e^{0.03 \times 3} = e^{0.09} \approx 1.0955$

Now that we have the value of the exponential expression, we can substitute it back into the original function to get:

$f(3) = 3 \times 1.0955 \approx 3.2865$

Rounding to the Nearest Tenth

Finally, we need to round the value of $f(3)$ to the nearest tenth. This gives us:

$f(3) \approx 3.3$

Conclusion

In this article, we have explored the function $f(t) = P e^{r t}$ and calculated the value of $f(3)$ given that $P = 3$ and $r = 0.03$. We have shown that the value of $f(3)$ is approximately $3.3$ when rounded to the nearest tenth.

Answer

The correct answer is B. 3.3.

Discussion

This problem is a classic example of an exponential function, which is commonly used in finance, physics, and other fields to model growth and decay. The function $f(t) = P e^{r t}$ is a type of exponential function, where $P$ is the initial value, $r$ is the growth rate, and $t$ is the time. In this case, we are given that $P = 3$ and $r = 0.03$. We need to calculate the value of $f(3)$, which means we need to find the value of the function when $t = 3$.

Related Topics

• Exponential functions
• Growth and decay
• Finance
• Physics

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Growth and Decay" by Khan Academy
• [3] "Finance" by Investopedia
• [4] "Physics" by Physics Classroom
  Q&A: Exponential Functions and Growth/Decay

Introduction

In our previous article, we explored the function $f(t) = P e^{r t}$ and calculated the value of $f(3)$ given that $P = 3$ and $r = 0.03$. In this article, we will answer some common questions related to exponential functions and growth/decay.

Q&A

Q: What is an exponential function?

A: An exponential function is a type of function that has the form $f(t) = P e^{r t}$, where $P$ is the initial value, $r$ is the growth rate, and $t$ is the time.

Q: What is the difference between exponential growth and exponential decay?

A: Exponential growth occurs when the value of the function increases over time, while exponential decay occurs when the value of the function decreases over time.

Q: How do you calculate the value of an exponential function?

A: To calculate the value of an exponential function, you need to substitute the given values into the function and evaluate the expression.

Q: What is the significance of the growth rate (r) in an exponential function?

A: The growth rate (r) determines the rate at which the value of the function increases or decreases over time.

Q: Can you give an example of an exponential function in real-life?

A: Yes, an example of an exponential function in real-life is the growth of a population over time. If the population of a city is increasing at a rate of 2% per year, the population can be modeled using an exponential function.

Q: How do you determine the initial value (P) in an exponential function?

A: The initial value (P) is the value of the function at time t = 0.

Q: Can you explain the concept of half-life in exponential decay?

A: Yes, the half-life of a substance is the time it takes for the substance to decay to half of its initial value. This concept is commonly used in chemistry and physics to model the decay of radioactive substances.

Q: How do you calculate the half-life of a substance?

A: To calculate the half-life of a substance, you need to use the formula: half-life = ln(2) / r, where r is the decay rate.

Conclusion

In this article, we have answered some common questions related to exponential functions and growth/decay. We hope that this article has provided you with a better understanding of these concepts and how they are used in real-life applications.

Related Topics

• Exponential functions
• Growth and decay
• Finance
• Physics
• Chemistry

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Growth and Decay" by Khan Academy
• [3] "Finance" by Investopedia
• [4] "Physics" by Physics Classroom
• [5] "Chemistry" by Chemistry LibreTexts

Further Reading

• "Exponential Functions: A Primer" by Wolfram MathWorld
• "Growth and Decay: A Tutorial" by MIT OpenCourseWare
• "Finance: A Guide to Exponential Functions" by Coursera
• "Physics: A Guide to Exponential Functions" by edX
• "Chemistry: A Guide to Exponential Functions" by Udemy