If $f(5)=288.9$ When $r=0.05$ For The Function $ F ( T ) = P E T F(t)=P E^t F ( T ) = P E T [/tex], Then What Is The Approximate Value Of $P$?A. 24 B. 3520 C. 225 D. 371

Mar 12, 2025 by ADMIN 179 views

**Solving for the Unknown Value of P in the Exponential Function**

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. We will use the function ${ f(t) = P e^{rt} }$ and the given value of ${ f(5) = 288.9 }$ when ${ r = 0.05 }$ to find the approximate value of ${ P }$ .

Understanding the Exponential Function

The exponential function is a mathematical function that describes how a quantity changes over time. It is commonly represented as ${ f(t) = P e^{rt} }$ , where ${ P }$ is the initial value, ${ r }$ is the growth rate, and ${ t }$ is the time.

Given Information

We are given the function ${ f(t) = P e^{rt} }$ and the value of ${ f(5) = 288.9 }$ when ${ r = 0.05 }$ . We need to find the approximate value of ${ P }$ .

Step 1: Substitute the Given Values into the Function

To find the value of ${ P }$ , we need to substitute the given values into the function. We have:

${ f(5) = P e^{r \cdot 5} }$

Substituting the given values, we get:

${ 288.9 = P e^{0.05 \cdot 5} }$

Step 2: Simplify the Equation

Now, we need to simplify the equation by evaluating the exponent:

${ 288.9 = P e^{0.25} }$

Step 3: Use the Properties of Exponents

We can use the properties of exponents to rewrite the equation:

${ 288.9 = P e^{0.25} }$

${ 288.9 = P \cdot e^{0.25} }$

Step 4: Evaluate the Exponent

To evaluate the exponent, we can use a calculator or a mathematical software:

${ e^{0.25} \approx 1.284 }$

Step 5: Substitute the Value of the Exponent

Now, we can substitute the value of the exponent into the equation:

${ 288.9 = P \cdot 1.284 }$

Step 6: Solve for P

To solve for ${ P }$ , we can divide both sides of the equation by 1.284:

${ P = \frac{288.9}{1.284} }$

Step 7: Calculate the Value of P

Now, we can calculate the value of ${ P }$ using a calculator or a mathematical software:

${ P \approx 225.5 }$

Conclusion

In this article, we used the function ${ f(t) = P e^{rt} }$ and the given value of ${ f(5) = 288.9 }$ when ${ r = 0.05 }$ to find the approximate value of ${ P }$ . We followed the steps outlined above and used the properties of exponents to simplify the equation. The approximate value of ${ P }$ is 225.5.

Answer

The approximate value of ${ P }$ is 225.5.

Comparison with the Given Options

The given options are:

A. 24 B. 3520 C. 225 D. 371

The approximate value of ${ P }$ we calculated is 225.5, which is closest to option C.

Final Answer

Introduction

In our previous article, we explored how to solve for an unknown value in a given exponential function. We used the function ${ f(t) = P e^{rt} }$ and the given value of ${ f(5) = 288.9 }$ when ${ r = 0.05 }$ to find the approximate value of ${ P }$ . In this article, we will answer some frequently asked questions related to solving for the unknown value of ${ P }$ in the exponential function.

Q: What is the exponential function?

A: The exponential function is a mathematical function that describes how a quantity changes over time. It is commonly represented as ${ f(t) = P e^{rt} }$ , where ${ P }$ is the initial value, ${ r }$ is the growth rate, and ${ t }$ is the time.

Q: How do I solve for the unknown value of P in the exponential function?

A: To solve for the unknown value of ${ P }$ , you need to substitute the given values into the function and use the properties of exponents to simplify the equation. You can then use a calculator or a mathematical software to evaluate the exponent and solve for ${ P }$ .

Q: What is the formula for the exponential function?

A: The formula for the exponential function is ${ f(t) = P e^{rt} }$ , where ${ P }$ is the initial value, ${ r }$ is the growth rate, and ${ t }$ is the time.

Q: How do I evaluate the exponent in the exponential function?

A: To evaluate the exponent, you can use a calculator or a mathematical software. You can also use the properties of exponents to rewrite the equation and simplify it.

Q: What is the approximate value of P in the given function?

A: The approximate value of ${ P }$ in the given function is 225.5.

Q: How do I compare the approximate value of P with the given options?

A: To compare the approximate value of ${ P }$ with the given options, you can use a calculator or a mathematical software to evaluate the options and determine which one is closest to the approximate value of ${ P }$ .

Q: What is the final answer?

A: The final answer is C.

Q: What are some common mistakes to avoid when solving for the unknown value of P in the exponential function?

A: Some common mistakes to avoid when solving for the unknown value of ${ P }$ in the exponential function include:

Not substituting the given values into the function
Not using the properties of exponents to simplify the equation
Not evaluating the exponent correctly
Not using a calculator or a mathematical software to evaluate the options

Conclusion

In this article, we answered some frequently asked questions related to solving for the unknown value of ${ P }$ in the exponential function. We provided step-by-step instructions on how to solve for ${ P }$ and highlighted some common mistakes to avoid. We also compared the approximate value of ${ P }$ with the given options and determined the final answer.

Additional Resources

For more information on solving for the unknown value of ${ P }$ in the exponential function, please refer to the following resources:

Final Answer

The final answer is C.