Solving For Time: B(t) = 1,280,000

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Hey guys! Today, we're diving into a math problem where we need to figure out a specific value of 't' that satisfies a given equation. It might sound intimidating, but trust me, we'll break it down step by step so it's super easy to follow. Our mission is to find the value of t, in days, that makes the equation b(t) = 1,280,000 true, using the formula b(t) = 409.6 ⋅ 5^(0.2t). Let's get started!

Understanding the Equation

First, let's make sure we understand what we're working with. The equation b(t) = 409.6 ⋅ 5^(0.2t) is a mathematical model. In this model, b(t) represents a value (in this case, 1,280,000) at a certain time t (which is what we want to find). The equation tells us how this value changes over time based on an exponential relationship. The number 409.6 is the initial value, 5 is the base of the exponent, and 0.2 is a constant that affects how quickly the value grows. Essentially, we're trying to find out how long it takes for b(t) to reach 1,280,000. This kind of problem pops up in various real-world scenarios, such as population growth, financial investments, and radioactive decay. By solving this equation, we're applying essential mathematical principles to understand and predict changes over time.

Key components of the equation:

  • b(t): The value at time t (in our case, 1,280,000).
  • 409.6: The initial value.
  • 5: The base of the exponent, indicating the rate of growth.
  • 0.2t: The exponent, which determines how the value changes over time.

Step-by-Step Solution

Alright, let's break down how to solve this equation. We'll go through each step in detail so you can follow along and understand exactly what we're doing.

Step 1: Set Up the Equation

Our first step is to set up the equation with the given values. We know that b(t) = 1,280,000, so we can write:

1,280,000 = 409.6 ⋅ 5^(0.2t)

This equation is now ready for us to solve for t.

Step 2: Isolate the Exponential Term

Next, we need to isolate the exponential term, which is 5^(0.2t). To do this, we'll divide both sides of the equation by 409.6:

1,280,000 / 409.6 = 5^(0.2t)

Calculating the left side, we get:

3125 = 5^(0.2t)

Step 3: Express Both Sides with the Same Base

To solve for t, it's helpful to express both sides of the equation with the same base. We know that 3125 is a power of 5. Specifically, 3125 = 5^5. So, we can rewrite the equation as:

5^5 = 5^(0.2t)

Step 4: Equate the Exponents

Now that both sides of the equation have the same base, we can equate the exponents:

5 = 0.2t

Step 5: Solve for t

Finally, we solve for t by dividing both sides of the equation by 0.2:

t = 5 / 0.2

t = 25

So, the value of t that satisfies the equation is 25 days.

Alternative Method: Using Logarithms

If expressing both sides with the same base isn't straightforward, we can use logarithms to solve for t. Here’s how:

Step 1: Start with the Isolated Exponential Term

As before, we start with the equation:

3125 = 5^(0.2t)

Step 2: Take the Logarithm of Both Sides

Take the logarithm of both sides. We can use any base for the logarithm, but the natural logarithm (ln) or the common logarithm (log base 10) are commonly used. Let’s use the natural logarithm:

ln(3125) = ln(5^(0.2t))

Step 3: Use Logarithm Properties to Simplify

Using the logarithm property ln(a^b) = b ⋅ ln(a), we can simplify the right side:

ln(3125) = 0.2t ⋅ ln(5)

Step 4: Solve for t

Now, solve for t by dividing both sides by 0.2 ⋅ ln(5):

t = ln(3125) / (0.2 ⋅ ln(5))

Using a calculator, we find:

ln(3125) ≈ 8.047

ln(5) ≈ 1.609

So,

t ≈ 8.047 / (0.2 ⋅ 1.609)

t ≈ 8.047 / 0.3218

t ≈ 25

Again, we find that the value of t is 25 days.

Verification

To make sure our answer is correct, let's plug t = 25 back into the original equation:

b(25) = 409.6 ⋅ 5^(0.2 * 25)

b(25) = 409.6 ⋅ 5^5

b(25) = 409.6 ⋅ 3125

b(25) = 1,280,000

Since the equation holds true, our answer is correct. t = 25 days is indeed the value that satisfies the equation b(t) = 1,280,000.

Practical Implications

Understanding how to solve equations like this is crucial in various real-world scenarios. For instance:

  • Finance: Calculating how long it takes for an investment to reach a certain value.
  • Biology: Modeling population growth over time.
  • Physics: Analyzing radioactive decay.
  • Business: Predicting sales growth based on current trends.

By mastering these mathematical techniques, you can make informed decisions and predictions in numerous fields. This isn't just about abstract math; it's about gaining practical insights into how things change and grow over time.

Conclusion

So, there you have it! We found that the value of t that satisfies the equation b(t) = 1,280,000, using the formula b(t) = 409.6 ⋅ 5^(0.2t), is 25 days. We walked through the steps to isolate the exponential term, express both sides with the same base, and solve for t. We also explored an alternative method using logarithms. Whether you're into finance, biology, or just love solving puzzles, understanding these concepts can be super useful. Keep practicing, and you'll become a pro at solving these types of equations in no time! Keep up the great work, guys! You've got this!