Problem 56 - Algebra 2

Question

The number of people \(N\) who will receive a forwarded e-mail can be approximated by \(N=\frac{P}{1+(P-S) e^{-0.35 t}},\) where \(P\) is the total number of people online, \(S\) is the number of people who start the e-mail, and \(t\) is the time in minutes. Suppose four people want to send an e-mail to all those who are online at that time. How much time will pass before half of the people will receive the e-mail?

Step-by-Step Solution

Answer

Time needed is \( t = -\frac{\ln\left(\frac{1}{P-4}\right)}{0.35} \).

Step 1: Understand the Given Equation

We start with the equation \( N = \frac{P}{1 + (P-S) e^{-0.35t}} \). This represents the number of people who receive the e-mail over time \( t \).

Step 2: Identify Key Parameters

Identify that \( P \) is the total number of people online, \( S \) is the number of people starting the e-mail, and \( t \) is time in minutes. In this problem, \( S = 4 \), as four people start the e-mail.

Step 3: Set the Problem Condition

We want to find the time \( t \) when half of the people online have received the e-mail. Thus, set \( N = \frac{P}{2} \).

Step 4: Substitute and Simplify

Substitute \( N = \frac{P}{2} \) into the equation, so \( \frac{P}{2} = \frac{P}{1 + (P-4) e^{-0.35t}} \). After substituting, multiply both sides by the denominator: \( P (1 + (P-4) e^{-0.35t}) = 2P \).

Step 5: Solve for the Exponential Term

Simplify to get \( P + P(P-4)e^{-0.35t} = 2P \), which reduces to \( P(P-4)e^{-0.35t} = P \). This simplifies to \( (P-4)e^{-0.35t} = 1 \).

Step 6: Isolate and Solve for Time

Solve the equation \( e^{-0.35t} = \frac{1}{P-4} \). Take the natural logarithm of both sides to get \( -0.35t = \ln\left(\frac{1}{P-4}\right) \). This simplifies to \( t = -\frac{\ln\left(\frac{1}{P-4}\right)}{0.35} \).

Key Concepts

Logarithmic Functions

Logarithmic functions may initially seem daunting, but they play a crucial role in solving many complex problems. A logarithm essentially tells us the power to which a number must be raised to obtain another number. In simpler terms, if you have a logarithm of base 10, \( \log_{10}(x) = y \), this means \( 10^y = x \). They are the inverse operations of exponentiation, much like how subtraction is the inverse of addition.

In our e-mail receiving exercise, we encounter the natural logarithm, denoted as \( \ln \), which uses the base \( e \), approximately equal to 2.71828. We use this in the solution to isolate the time \( t \) in an exponential equation involving the natural number \( e^{-0.35t} \). By taking the natural logarithm, we simplify our calculations to solve for \( t \), which provides us with a clearer understanding of how exponential growth affects the number of e-mail recipients over time.

Algebraic Equations

An algebraic equation is a mathematical sentence that uses numbers, variables, and operation symbols. It often represents a balance, with an expression on each side of an equal sign. In the given exercise, we use algebraic equations to determine when half of the online population will receive a given e-mail.

Our primary equation is \( N = \frac{P}{1 + (P-S)e^{-0.35t}} \), where each letter or symbol in the equation has a specific meaning or function. Here, \( P \) is the number of people online, \( S \) is the seed number starting the e-mail, and \( e^{-0.35t} \) is our exponential factor. This equation helps us break down the complex relationship between time and the diffusion of information, leading us to solve for the special condition where half the online populace has received the e-mail.

Problem-Solving in Algebra

Problem-solving in algebra involves a systematic approach to find unknown values by using known information, laws, and algebraic manipulation. This involves several steps:

• Understand what the problem is asking. For instance, calculating the time for half of the online people to receive an e-mail.
• Identify known variables and their values. Here, we know \( S = 4 \) and wish to find the corresponding \( t \).
• Set up equations using the known parameters to express the unknown. We expressed the condition \( N = \frac{P}{2} \).
• Simplify the equation to isolate the unknown variable. We achieved this by multiplying both sides and simplifying to find \( t \).
• Apply the natural logarithm for isolating exponential expressions. This is crucial since it converts the exponential equation into a linear one that is easier to solve.

This detailed, step-by-step method helps solve numerous algebra problems, allowing us to convert an unknown into a known through logical and mathematical reasoning.