Problem 60 - Elementary Algebra

Question

The cost of producing a subscription website consists of an initial programming and setup fee of \(\$ 4,500\) plus a monthly Web hosting fee of \(\$ 29.95\). The cost of creating and hosting the website can be calculated using the formula \(C=4500+29.95 n\), where \(n\) represents the number of months the website is hosted. How much will it cost to set up and host the website for 1 year?

Step-by-Step Solution

Answer

The cost for 1 year is $4859.40.

Step 1: Understand the Problem

The problem gives us a formula to calculate the cost of creating and hosting a website: \(C = 4500 + 29.95n\). Here, \(n\) represents the number of months the website is hosted. We're asked to find the cost for 1 year.

Step 2: Convert Years to Months

Since \(n\) is in months, we need to convert 1 year to months. There are 12 months in a year, so \(n = 12\).

Step 3: Substitute the Value of n

Now that we have \(n = 12\), substitute this value into the formula: \(C = 4500 + 29.95(12)\).

Step 4: Calculate the Monthly Hosting Cost

Perform the multiplication to find the total cost of hosting for 12 months: \(29.95 \times 12 = 359.40\).

Step 5: Calculate the Total Cost

Add the initial fee to the total hosting cost: \(4500 + 359.40 = 4859.40\). Thus, the total cost for setting up and hosting the website for 1 year is \(\$4859.40\).

Key Concepts

Cost Calculation in Everyday Problems

Cost calculation is an essential skill in elementary algebra that helps solve everyday problems, such as determining the expenses of a subscription service. By understanding and working with cost formulas, you can predict future costs and make budgetary decisions.

In the given problem, the formula is:

• \( C = 4500 + 29.95n \)

This formula calculates the total cost \( C \) of setting up and hosting a website.
The initial fixed cost (programming and setup fee) is \( \\( 4500 \), and the monthly recurring cost (hosting) is \( \\) 29.95 \) per month.

To find a total cost for multiple months, you must sum both the fixed and variable costs. Avoid errors by double-checking your calculations and substituting the correct values into the formula. This method ensures a clear understanding of how costs are distributed over time, allowing for better financial planning.

Understanding Linear Equations

Linear equations form the foundation of many algebraic problems and are often used in cost calculations. A linear equation is an equation of the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.

In the exercise at hand, the equation \( C = 4500 + 29.95n \) is a linear equation, as it represents a straight line when plotted on a graph.
This form illustrates how costs are structured based on the number of months, \( n \).
The term "4500" is the y-intercept, representing a fixed starting cost.
Meanwhile, "29.95n" is the slope, indicating how much the total cost increases for each additional month of hosting.

By understanding the components of a linear equation, students can analyze how changes in one variable affect the outcome. This understanding allows you to solve similar cost problems by adjusting the variables according to different scenarios.

Using Mathematical Formulas

Mathematical formulas provide a set methodology for solving problems, ensuring consistent and reliable results.
In this exercise, we use the formula \( C = 4500 + 29.95n \).

Here's how it helps:

• Defines the relationship between costs and time by specifying both fixed and variable costs.
• Offers a straightforward method to calculate costs for any number of months once \( n \) is known.
• Simplifies complex information into an easily understood format.

The use of formulas like this one allows you to substitute different values for the variable \( n \), enabling flexible calculations across varied scenarios.
Such flexibility ensures that students can adapt their approach to solve a wide range of cost-related problems, reinforcing algebraic proficiency and bolstering problem-solving skills.