Problem 75 - Precalculus : Building Concepts and Connections

Question

A wireless phone company has a pricing scheme that includes 250 minutes worth of phone usage in the basic monthly fee of \(\$ 30 .\) For each minute over and above the first 250 minutes of usage, the user is charged an additional \(\$ 0.60\) per minute. (a) Let \(x\) be the number of minutes of phone usage per month. What is the expression for the average cost per minute if the value of \(x\) is in the interval (0,250)\(?\) (b) What is the expression for the average cost per minute if the value of \(x\) is above \(250 ?\) (c) If phone usage in a certain month is 600 minutes, what is the average cost per minute?

Step-by-Step Solution

Answer

The average cost per minute when usage is within the 250 minutes mark is \(\frac{30}{x}\), when usage exceeds 250 minutes it's \(\frac{30 + 0.60(x - 250)}{x}\), and for 600 minutes of usage it is \(\frac{30 + 0.60(600 - 250)}{600}\).

Step 1: Calculating average cost per minute when usage is within the 250 minutes mark

Given that the total cost for 250 minutes is $30, we can calculate the average cost per minute as the total cost divided by the number of minutes. Therefore, the expression for the average cost per minute when the usage is within the 250 minutes mark is \(\frac{30}{x}\), where \(x\) is the number of minutes used.

Step 2: Calculating average cost per minute when usage exceeds the 250 minutes mark

When usage exceeds 250 minutes, for each additional minute, a charge of $0.60 is added to the flat fee of $30. This adds to the total cost. Therefore, the average cost per minute is the total cost divided by the total number of minutes used. The total cost in this case is $30 for the first 250 minutes and $0.60 for each additional minute (i.e., \(x - 250\)). Therefore, the expression for the average cost per minute when the usage exceeds 250 minutes is \(\frac{30 + 0.60(x - 250)}{x}\), where \(x\) is the total number of minutes used.

Step 3: Calculating average cost per minute for 600 minutes usage

For this part, we plug \(x = 600\) into our equation from step 2. So it becomes \(\frac{30 + 0.60(600 - 250)}{600}\). This gives us the average cost per minute for 600 minutes of usage.

Key Concepts

Understanding Wireless Phone Pricing

Wireless phone pricing can sometimes appear complex due to different factors that influence the total cost. When dealing with companies that offer wireless phone services, it's essential to understand how they structure their pricing. Many plans include a basic monthly fee that covers a set amount of usage, in this case, 250 minutes for $30. Here’s a simplified look at how this particular plan works:

• The basic plan has a fixed cost of $30 per month.
• This fee covers up to 250 minutes of phone usage.
• If you use fewer than 250 minutes, your per-minute cost decreases as you use more minutes."
• Once you exceed 250 minutes, additional charges apply for each extra minute you talk.

This pricing method allows users to pay a predictable fee unless they exceed the given minutes, which then prompts additional charges.

Explaining the Basic Monthly Fee

The basic monthly fee is the cornerstone of many wireless phone service pricing models. It's essentially a fixed cost that provides a certain package of services or allowances. In this instance, the wireless phone plan's basic monthly fee is \(30, covering 250 minutes of talk time. This fee provides a flat rate that makes budgeting easier for users. Here’s a breakdown of its components:

• This fee is payable regardless of how many minutes are used, as long as usage remains at or below 250 minutes.
• If a user utilizes exactly 250 minutes, the cost per minute stands at \)\frac{30}{250}\( or \)0.12 per minute.
• For usage under 250 minutes, the effective cost per minute becomes lower with each additional minute used up to the 250-minute threshold.

The basic monthly fee thus acts as a safety net, ensuring that users pay a predictable monthly amount for a defined level of service.

Additional Charge Calculation for Exceeding Minutes

When a user exceeds the 250 included minutes in the basic monthly fee, they incur additional charges. Understanding how these additional costs are calculated is crucial to managing one's budget.

• For every minute over the 250-minute limit, an additional charge of \(0.60 per minute is applied."
• To find the total cost when usage exceeds 250 minutes, you must add these extra charges to the basic monthly fee.

For example, if the total usage is 600 minutes, the additional minutes used are 600 - 250 = 350 minutes. Each of these 350 additional minutes costs \)0.60, so the total additional cost is \(0.60 \times 350. Combined with the \)30 basic fee, the total cost for the month is 30 + (0.60 \times 350).This understanding allows users to calculate their monthly expenses more accurately, noting that exceeding included minutes will significantly impact the per-minute cost.