Modeling Blog Growth Determining Time To Reach 6575 Views

In this article, we will explore how to solve a problem involving exponential growth, specifically in the context of modeling blog views over time. Exponential growth models are commonly used to describe phenomena that increase rapidly over time, such as population growth, compound interest, and, as in this case, the number of views on an educational blog. The key to solving these problems lies in understanding the underlying equation and using algebraic techniques to isolate the variable we are interested in. Let's dive into the problem and break down the steps to find the solution.

The Problem: Modeling Educational Blog Views

The problem at hand involves the function $\bf{P(t) = \frac{10,000}{1 + 9e^{-0.0625t}}}$, which models the total number of views an educational blog has received after $t$ minutes. Our goal is to determine the time, $t$, when the blog will have reached a total of 6,575 views. This type of problem is a classic example of applying mathematical models to real-world scenarios. Understanding how to manipulate and solve such equations is a valuable skill in various fields, including business, science, and technology. The equation itself represents a logistic growth model, which is a type of exponential model that takes into account a carrying capacity—in this case, the maximum number of views the blog can potentially receive. The numerator, 10,000, represents this carrying capacity, while the denominator incorporates an exponential decay term that influences the rate at which the views approach this limit. By setting $P(t)$ equal to 6,575 and solving for $t$, we can pinpoint the specific time when the blog reaches this milestone. This process involves several algebraic steps, including isolating the exponential term, taking the natural logarithm of both sides, and finally, solving for $t$. Let's go through these steps methodically to arrive at the correct answer.

Setting Up the Equation

The core of solving this problem is to set the function $P(t)$ equal to the desired number of views, which is 6,575. This gives us the equation:

$\bf{6,575 = \frac{10,000}{1 + 9e^{-0.0625t}}}$

This equation represents the point at which the total number of views, as predicted by our model, reaches the target of 6,575. The next step is to isolate the exponential term, $e^{-0.0625t}$, which contains the variable we want to solve for, $t$. This involves a series of algebraic manipulations to gradually simplify the equation and bring us closer to finding the value of $t$. The process of isolating the exponential term is crucial in solving exponential equations, as it allows us to apply the inverse operation—the natural logarithm—to both sides of the equation. This, in turn, will help us to eliminate the exponential and bring the variable $t$ out of the exponent, making it solvable. The initial step in this isolation process typically involves multiplying both sides of the equation by the denominator of the fraction, which in this case is $(1 + 9e^{-0.0625t})$. This will eliminate the fraction and leave us with a more manageable equation to work with. Then, we can proceed with further algebraic steps to isolate the exponential term completely. Let's move on to the next step and see how we can continue to simplify the equation.

Isolating the Exponential Term

To isolate the exponential term, we first multiply both sides of the equation by $(1 + 9e^{-0.0625t})$:

$\bf{6,575(1 + 9e^{-0.0625t}) = 10,000}$

Next, distribute the 6,575 on the left side:

$\bf{6,575 + 59,175e^{-0.0625t} = 10,000}$

Now, subtract 6,575 from both sides:

$\bf{59,175e^{-0.0625t} = 3,425}$

Finally, divide both sides by 59,175:

$\bf{e^{-0.0625t} = \frac{3,425}{59,175} ≈ 0.0579}$

This step is crucial because it brings us to a point where the exponential term is by itself on one side of the equation. This allows us to use the natural logarithm, the inverse function of the exponential function, to eliminate the exponential and solve for $t$. The isolation process involves a series of algebraic manipulations, each designed to simplify the equation and bring the exponential term closer to being isolated. These manipulations include multiplying both sides by a common factor, distributing terms, subtracting constants, and dividing by coefficients. Each of these steps must be performed carefully to maintain the equality of the equation and avoid introducing errors. Once the exponential term is isolated, we are ready to take the natural logarithm of both sides, which will be the key to unlocking the value of $t$. Let's proceed to the next step and see how the natural logarithm helps us solve the equation.

Applying the Natural Logarithm

To solve for $t$, we take the natural logarithm (ln) of both sides of the equation:

$\bf{ln(e^{-0.0625t}) = ln(0.0579)}$

Using the property of logarithms that $ln(e^x) = x$, we get:

$\bf{-0.0625t = ln(0.0579)}$

The natural logarithm is the inverse function of the exponential function with base $e$, and it is a powerful tool for solving exponential equations. By taking the natural logarithm of both sides, we effectively undo the exponential operation and bring the exponent down as a coefficient. This is the key step in isolating the variable $t$, as it removes it from the exponent and allows us to solve for it directly. The property of logarithms that allows us to simplify $ln(e^x)$ to $x$ is a fundamental property that is essential for solving exponential equations. Understanding and applying this property correctly is crucial for arriving at the correct solution. In this case, applying the natural logarithm simplifies our equation to a linear equation in terms of $t$, which is much easier to solve. The next step is simply to divide both sides by the coefficient of $t$ to find the value of $t$. Let's move on to the final steps and calculate the value of $t$.

Solving for t

Now, we divide both sides by -0.0625:

$\bf{t = \frac{ln(0.0579)}{-0.0625}}$

Using a calculator, we find:

$\bf{t ≈ \frac{-2.849}{-0.0625} ≈ 45.584}$

Rounding to one decimal place, we get $t ≈ 45.6$ minutes.

This final step involves a simple division to isolate $t$ and obtain its value. The value of $t$ represents the number of minutes it will take for the blog to reach 6,575 views, according to our model. It is important to remember to round the answer appropriately, based on the context of the problem and the level of precision required. In this case, rounding to one decimal place seems reasonable, as it provides a balance between accuracy and readability. The result, $t ≈ 45.6$ minutes, gives us a concrete answer to our original question. It tells us that it will take approximately 45.6 minutes for the blog to reach the milestone of 6,575 views. This information can be valuable for the blog's creators, as it allows them to track their progress and plan their content strategy accordingly. Let's summarize our solution and consider the implications of our findings.

Conclusion: The Blog Reaches 6,575 Views at Approximately 45.6 Minutes

After working through the steps, we've determined that the blog will have approximately 6,575 total views after about 45.6 minutes. Therefore, none of the provided answer choices (A. 20.4 minutes, B. 36.1 minutes, C. 39.1 minutes) are correct. The correct answer is approximately 45.6 minutes.

This exercise demonstrates the power of mathematical models in predicting real-world phenomena. By understanding the equation that governs the growth of blog views, we can accurately estimate when the blog will reach certain milestones. This type of analysis is valuable in various fields, from marketing and business to science and engineering. The process of solving exponential equations involves a series of algebraic manipulations, including isolating the exponential term, taking the natural logarithm, and solving for the variable of interest. Each step must be performed carefully to ensure accuracy and avoid errors. The final result, $t ≈ 45.6$ minutes, provides a concrete answer to our original question and highlights the importance of mathematical modeling in understanding and predicting real-world events. This problem also reinforces the importance of carefully reviewing answer choices and ensuring that they align with our calculated result. In this case, none of the provided answer choices were correct, emphasizing the need for thoroughness and attention to detail in problem-solving. The skills and techniques we've used in this article can be applied to a wide range of problems involving exponential growth and decay, making them valuable tools for anyone interested in understanding and predicting change over time.