Question

Spam Messages The total number of email messages per day (in billions) between 2003 and 2007 is approximated by $$ f(t)=1.54 t^{2}+7.1 t+31.4 \quad 0 \leq t \leq 4 $$ where \(t\) is measured in years, with \(t=0\) corresponding to 2003\. Over the same period the total number of spam messages per day (in billions) is approximated by $$ g(t)=1.21 t^{2}+6 t+14.5 \quad 0 \leq t \leq 4 $$ a. Find the rule for the function \(D=f-g .\) Compute \(D(4)\), and explain what it measures. b. Find the rule for the function \(P=g / f\). Compute \(P(4)\), and explain what it means.

Short answer

In 2007, the difference between the total number of email messages and the number of spam messages per day in billions was 37.94 billion. Additionally, about 53.21% of the total email messages per day were spam messages in 2007.

Step-by-step solution

Calculate the difference between f(t) and g(t)

We need to find the difference between the two functions, \(f(t)\) and \(g(t)\), and we will denote this difference as \(D(t)\). So, we simply subtract \(g(t)\) from \(f(t)\). $$ D(t) = f(t) - g(t) $$

Plug in the expressions of f(t) and g(t) and simplify

We found that \(D(t) = f(t) - g(t)\). Now, we just need to plug in the given expressions of \(f(t)\) and \(g(t)\) and simplify the result: $$ D(t) = (1.54 t^{2}+7.1 t+31.4) - (1.21 t^{2}+6 t+14.5) $$ Simplify: $$ D(t) = (1.54 t^{2} - 1.21 t^{2}) + (7.1 t - 6 t) + (31.4 - 14.5) $$ $$ D(t) = 0.33t^{2} + 1.1t + 16.9 $$

Compute D(4) and explain its significance

Now, we need to compute the value of \(D(t)\) at \(t = 4\): $$ D(4) = 0.33(4)^{2} + 1.1(4) + 16.9 $$ $$ D(4) = 16.64 + 4.4 + 16.9 = 37.94 \ (\text{billion}) $$ \(D(4)\), which is 37.94 billion, represents the difference between the total number of email messages and the number of spam messages per day in billions in 2007. #b. Finding P(t)=g(t)/f(t)#

Calculate the ratio between g(t) and f(t)

We need to find the ratio between the two functions, \(g(t)\) and \(f(t)\), and we will denote this ratio as \(P(t)\). So, we simply divide \(g(t)\) by \(f(t)\): $$ P(t) = \frac{g(t)}{f(t)} $$

Plug in the expressions of g(t) and f(t)

We found that \(P(t) = \frac{g(t)}{f(t)}\). Now, we just need to plug in the given expressions of \(g(t)\) and \(f(t)\) into the ratio: $$ P(t) = \frac{1.21 t^{2}+6 t+14.5}{1.54 t^{2}+7.1 t+31.4} $$

Compute P(4) and explain its meaning

Now, we need to compute the value of \(P(t)\) at \(t = 4\): $$ P(4) = \frac{1.21(4)^{2} + 6(4) + 14.5}{1.54(4)^{2} + 7.1(4) + 31.4} $$ $$ P(4) = \frac{77.8}{146.28} \approx 0.5321 $$ \(P(4)\) is approximately 0.5321, which means that in 2007, about 53.21% of the total email messages per day were spam messages.

Key concepts

Difference of Functions

In calculus, when we talk about the difference of functions, we are referring to the subtraction of one function from another. This operation helps us understand how one quantity changes relative to another. For instance, in the email and spam message problem, we are given two functions: f(t) representing the total number of email messages and g(t) representing the spam messages over time. The difference D(t) = f(t) - g(t) provides us with a new function that quantifies the disparity between these two values at any given time t.

Computing D(t) for a specific value, such as t = 4 (representing the year 2007), yields an actual number that tells us how many more total emails were sent compared to spam emails. This result is significant as it sheds light on the magnitude of spam in overall email traffic during that year. In our example, finding that D(4) = 37.94 billion emails clarifies that this is the excess number of non-spam emails over spam emails, illustrating the difference function's practical utility in real-world problems.

Function Ratio

The function ratio involves dividing one function by another. This is expressed as P(t) = g(t) / f(t), where P(t) captures the proportion of one function relative to the other function at time t. In our case, it represents the ratio of spam messages to total email messages. Analyzing the function ratio is crucial as it provides insights into the relative behavior of two quantities. For example, by computing P(4), we learn that approximately 53.21% of all emails were spam in 2007.

This ratio can reveal trends, such as an increasing prevalence of spam over the years if the percentage grows with time. It's an invaluable tool for statisticians and analysts who want to make sense of how one aspect of a dataset relates to the whole. Moreover, in different contexts, function ratios can assist in comparing rates of change, which is a key concept in differential calculus.

Real-world Application of Calculus

The application of calculus in the real world is vast and diverse. The email and spam message calculus problem exemplifies how calculus can model complex real-life situations. Through functions, one can simulate realistic phenomena such as the growth of email traffic over time. The ability to find the function difference and ratio enables us to extract meaningful information from these models—such as the excess number of non-spam emails or the percentage of spam emails in relation to the total.

These calculus tools help in making informed decisions, whether by internet service providers managing their networks, businesses shaping their email marketing strategies, or policymakers understanding the scope of digital communication challenges. The power of calculus to describe and predict patterns in a wide range of fields like economics, engineering, and environmental science is why it's a foundational element in the toolkit of problem-solvers across industries.