Calculate F(5) For F(t) = Pe^t With P = 5 And R = 0.05

Hey guys! Let's dive into this math problem where we need to find the value of an exponential function. We've got the function f(t) = Pe^t, and we're given that P = 5 and r = 0.05. Our mission? To calculate f(5) and round it to the nearest tenth. This might seem a bit daunting at first, but don't worry, we'll break it down step by step. Exponential functions are super important in many fields, from finance to biology, so understanding how to work with them is a valuable skill. We will walk you through each step, ensuring that you understand not just the solution, but also the process of arriving at the answer. So, grab your calculators and let’s get started! 🚀

Understanding the Exponential Function

Before we jump into the calculation, let's make sure we're all on the same page about what an exponential function is. The function f(t) = Pe^t is a classic example of an exponential function. In this equation:

• f(t) is the value of the function at time t.
• P is the initial value or principal amount.
• e is Euler's number, which is approximately equal to 2.71828.
• t is the time variable.

In our specific problem, we have P = 5, which means our initial value is 5. The e is a constant, and t is the variable we'll plug in to find the function's value at that time. The 'r' which equals 0.05, often represents the rate of growth or decay in exponential models. However, in the equation provided (f(t) = Pe^t), 'r' doesn't explicitly appear. It's possible there was a slight misunderstanding or typo in the problem statement, or it might be implied within a larger context that isn't provided here. For the purpose of solving the given equation, we'll proceed without directly using the value of 'r'. The exponential function is a powerful tool for modeling various phenomena, such as population growth, radioactive decay, and compound interest. It's characterized by rapid growth or decay, depending on the base of the exponent. This rapid change is what makes exponential functions so useful in modeling real-world scenarios where quantities change drastically over time. So, with a solid grasp of what our function represents, let's move on to the calculation part.

Plugging in the Values

Okay, now for the fun part – plugging in the values! We know that f(t) = Pe^t, P = 5, and we want to find f(5). So, we substitute t = 5 into the equation. This gives us:

f(5) = 5 * e^(5*0.05)

Notice how we've replaced t with 5 in the equation. We also include the value of r (0.05) in the exponent, since the function actually uses e^(rt). This is a crucial step, as it sets the stage for us to calculate the value of the function at the specific time we're interested in. Now, let's break down the exponent. We have 5 multiplied by 0.05, which equals 0.25. So our equation now looks like this:

f(5) = 5 * e^(0.25)

This simplifies our calculation quite a bit. We've gone from a function with a variable t to a specific calculation that we can solve using a calculator. The next step involves evaluating e^(0.25). Remember, e is Euler's number, approximately 2.71828, so e^(0.25) means we're raising this number to the power of 0.25. This calculation is best done with a calculator that has an exponential function, often labeled as e^x. Once we find the value of e^(0.25), we'll simply multiply it by 5 to get our final answer. So, let’s move on to the next step and evaluate that exponential part!

Evaluating e^(0.25)

Alright, let's tackle e^(0.25). You'll need a calculator with an exponential function (usually an "e^x" button) for this step. Inputting e^(0.25) into a calculator, we get approximately:

e^(0.25) ≈ 1.28402541669

Wow, that's a lot of digits! But don't worry, we're not going to use them all. For the sake of accuracy in our final answer, it's good to keep a few extra decimal places during the calculation. Now that we have the value of e^(0.25), we're just one step away from finding f(5). We've already simplified our equation to f(5) = 5 * e^(0.25), so we just need to multiply this value by 5. This step is pretty straightforward, and it brings us closer to our final answer, which we'll then round to the nearest tenth as the problem asks. So, let's go ahead and do that multiplication in the next section!

Multiplying by 5

Now that we know e^(0.25) is approximately 1.28402541669, we can plug that back into our equation:

f(5) = 5 * 1.28402541669

Multiplying 5 by 1.28402541669, we get:

f(5) ≈ 6.42012708345

We're almost there! We've calculated the value of f(5), but the question asks us to round it to the nearest tenth. This means we need to look at the digit in the hundredths place (the second digit after the decimal) to determine whether to round up or down. In our case, the digit in the hundredths place is 2, which is less than 5, so we'll round down. This final rounding step ensures that our answer is in the format requested by the problem. Rounding is a crucial skill in mathematics and sciences, as it allows us to simplify results and present them in a meaningful way. So, let’s do that final rounding and get our answer!

Rounding to the Nearest Tenth

Okay, we've got f(5) ≈ 6.42012708345, and we need to round this to the nearest tenth. Looking at the hundredths place, we see a 2. Since 2 is less than 5, we round down. This means we keep the digit in the tenths place (which is 4) as it is.

So, rounding 6.42012708345 to the nearest tenth gives us:

f(5) ≈ 6.4

And there we have it! We've successfully calculated f(5) and rounded it to the nearest tenth. This entire process has taken us from understanding the exponential function, plugging in the values, evaluating the exponential term, multiplying, and finally, rounding to get our answer. Each step is crucial, and mastering these steps will help you tackle similar problems with confidence. So, the value of f(5) to the nearest tenth is 6.4. Great job, guys! Let's celebrate this small victory! 🎉

Therefore, the answer is A. 6.4