If $f(3)=191.5$ When $r=0.03$ For The Function \$f(t)=P E^t$[/tex\], Then What Is The Approximate Value Of $P$?A. 175 B. 471 C. 210 D. 78

Introduction

In this article, we will delve into the world of exponential functions and explore how to solve for an unknown value in a given function. Specifically, we will be working with the function ${ f(t) = P e^{rt} }$, where ${ P }$ is the initial value, ${ r }$ is the growth rate, and ${ t }$ is the time. We will use the given information that ${ f(3) = 191.5 }$ when ${ r = 0.03 }$ to find the approximate value of ${ P }$.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes how a quantity changes over time. The general form of an exponential function is ${ f(t) = a e^{bt} }$, where ${ a }$ is the initial value, ${ b }$ is the growth rate, and ${ t }$ is the time. In our case, the function is ${ f(t) = P e^{rt} }$, where ${ P }$ is the initial value, ${ r }$ is the growth rate, and ${ t }$ is the time.

Given Information

We are given that ${ f(3) = 191.5 }$ when ${ r = 0.03 }$. This means that when the time ${ t }$ is 3, the value of the function ${ f(t) }$ is 191.5, and the growth rate ${ r }$ is 0.03.

Solving for P

To solve for ${ P }$, we can use the given information and the definition of the exponential function. We know that ${ f(3) = 191.5 }$ when ${ r = 0.03 }$, so we can substitute these values into the function:

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

Simplifying the equation, we get:

${ 191.5 = P e^{0.09} }$

To solve for ${ P }$, we can divide both sides of the equation by ${ e^{0.09} }$:

${ P = \frac{191.5}{e^{0.09}} }$

Using a calculator to evaluate the expression, we get:

${ P \approx 175.32 }$

Rounding to the nearest whole number, we get:

${ P \approx 175 }$

Conclusion

In this article, we used the given information that ${ f(3) = 191.5 }$ when ${ r = 0.03 }$ to find the approximate value of ${ P }$ in the function ${ f(t) = P e^{rt} }$. We solved for ${ P }$ by substituting the given values into the function and using algebraic manipulations to isolate ${ P }$. The approximate value of ${ P }$ is 175.

Answer

The approximate value of ${ P }$ is 175.

Discussion

This problem is a classic example of how to solve for an unknown value in an exponential function. The key concept is to use the given information and the definition of the exponential function to set up an equation, and then use algebraic manipulations to solve for the unknown value. This type of problem is commonly encountered in mathematics and science, and is an important tool for modeling real-world phenomena.

Related Topics

• Exponential functions
• Algebraic manipulations
• Modeling real-world phenomena

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Algebraic Manipulations" by Khan Academy
• [3] "Modeling Real-World Phenomena" by Wolfram MathWorld
  Q&A: Solving for P in an Exponential Function =====================================================

Introduction

In our previous article, we explored how to solve for the unknown value of P in the exponential function ${ f(t) = P e^{rt} }$. We used the given information that ${ f(3) = 191.5 }$ when ${ r = 0.03 }$ to find the approximate value of P. In this article, we will answer some common questions related to solving for P in an exponential function.

Q: What is the formula for solving for P in an exponential function?

A: The formula for solving for P in an exponential function is:

${ P = \frac{f(t)}{e^{rt}} }$

Where ${ f(t) }$ is the value of the function at time t, ${ r }$ is the growth rate, and ${ t }$ is the time.

Q: How do I know if the value of P is positive or negative?

A: To determine if the value of P is positive or negative, you need to examine the signs of the values in the equation. If the value of ${ f(t) }$ is positive and the value of ${ e^{rt} }$ is positive, then the value of P will be positive. If the value of ${ f(t) }$ is negative and the value of ${ e^{rt} }$ is positive, then the value of P will be negative.

Q: What if the value of r is negative? How does that affect the value of P?

A: If the value of r is negative, then the value of ${ e^{rt} }$ will be less than 1, and the value of P will be greater than ${ f(t) }$. This is because the exponential function with a negative exponent is less than 1.

Q: Can I use this method to solve for P in any exponential function?

A: Yes, you can use this method to solve for P in any exponential function of the form ${ f(t) = P e^{rt} }$. However, you need to make sure that the value of r is not zero, because in that case the function would be linear, not exponential.

Q: What if I have multiple values of f(t) and r? Can I still use this method?

A: Yes, you can use this method even if you have multiple values of f(t) and r. You just need to substitute each value of f(t) and r into the equation and solve for P separately.

Q: Are there any other ways to solve for P in an exponential function?

A: Yes, there are other ways to solve for P in an exponential function. One way is to use logarithms to rewrite the equation in a form that is easier to solve. Another way is to use numerical methods, such as the Newton-Raphson method, to approximate the value of P.

Conclusion

In this article, we answered some common questions related to solving for P in an exponential function. We covered topics such as the formula for solving for P, how to determine the sign of P, and how to handle negative values of r. We also discussed the limitations of this method and alternative methods for solving for P.

Related Topics

• Exponential functions
• Algebraic manipulations
• Modeling real-world phenomena
• Logarithms
• Numerical methods

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Algebraic Manipulations" by Khan Academy
• [3] "Modeling Real-World Phenomena" by Wolfram MathWorld
• [4] "Logarithms" by Math Is Fun
• [5] "Numerical Methods" by MIT OpenCourseWare