Problem 46 - Algebra 2

Question

A Webmaster estimates that the time (seconds) to connect to the server when \(n\) people are connecting is given by \(t(n)=0.005 n+0.3 .\) Estimate the time to connect when 50 people are connecting.

Step-by-Step Solution

Answer

The estimated time to connect with 50 people is 0.55 seconds.

Step 1: Understand the Problem

The problem gives us a formula for the time it takes to connect to a server based on the number of people, \(n\), connecting. We are asked to find the time when 50 people are connecting.

Step 2: Substitute the Given Value

Substitute \(n = 50\) into the given formula \(t(n) = 0.005n + 0.3\). This will allow us to calculate the specific time for 50 people connecting.

Step 3: Perform the Calculation

Now calculate the expression: \(t(50) = 0.005 \times 50 + 0.3\). Compute each part: \(0.005 \times 50 = 0.25\), then add \(0.3\), resulting in \(0.55\).

Step 4: State the Result

After calculating, the estimated time to connect when 50 people are connecting is \(0.55\) seconds.

Key Concepts

Linear Equations

Linear equations are fundamental mathematical expressions that describe a straight line when plotted on a graph. They typically follow the form \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. In our example related to the server connection time, the equation given is \( t(n) = 0.005n + 0.3 \).

Here, \( t(n) \) represents the time it takes to connect, \( n \) is the number of people connecting, \( 0.005 \) represents the rate at which connection time increases per person, and \( 0.3 \) is the base connection time when no one is connecting.

• Understand what each term in the equation means.
• Notice how the equation describes the change in connection time.
• This understanding will help you apply the concept in various similar real-life situations.

By interpreting these values, you grasp how a linear equation models the time relationship as more people connect to a server.

Substitution Method

The substitution method is a handy tool for solving equations, especially when you have specific values to plug into a formula. The aim is to simplify the calculation by replacing a variable with a given number, making it easier to find the solution.

To apply this method here, when asked to estimate the connection time for 50 people, follow these steps:

• Identify the variable in the equation, which is \( n \).
• Substitute this variable with the number 50, since our task is to find the connection time for precisely 50 users.
• Use the formula \( t(n) = 0.005n + 0.3 \). Plug \( n = 50 \) into it, resulting in \( t(50) = 0.005 \times 50 + 0.3 \).

Through substitution, you break down potentially complex problems into easier, straight-forward calculations.

Problem Solving

Problem solving in mathematics often involves several steps, from understanding the problem to generating a solution. It's important to approach each step carefully and methodically to ensure accuracy.

• Start by thoroughly reading and comprehending the problem. If it gives a formula or equation, identify each part.
• Decide on the method to use for finding the solution, like using substitution in our example.
• Apply the chosen method to the problem, execute the calculations, and finally verify your result.

In our connection time problem, we:

• Identified the problem asking for a specific calculation.
• Used substitution to find the precise connection time for 50 people.
• Followed through step-by-step until arriving at \(0.55\) seconds.

Learning these steps builds confidence and adds a structured strategy for tackling a variety of challenges, not just mathematical ones.