Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

A telephone company charges customers a certain amount for the first minute of a long distance call and another amount for each additional minute. A customer makes two calls to the same citya 36 -minute call for $$\$ 2.93$$ and a 13-minute call for $$\$ 1.09$$. (a) Determine the cost for the first minute and the cost for each additional minute. (b) If there is a federal tax rate of \(3.2 \%\) and a state tax rate of \(7.2 \%\) on all long distance calls, find, to the nearest minute, the longest call to the same city whose cost will not exceed \$5.00.

Step-by-Step Solution

Answer

First minute costs $0.13; each additional minute costs $0.08. Max call length with tax is 56 mins.

Step 1: Set Up Equations

Let's denote the cost for the first minute as \( x \) and the cost for each additional minute as \( y \). For a 36-minute call costing \( \\(2.93 \), the equation is:\[ x + 35y = 2.93 \]For a 13-minute call costing \( \\)1.09 \), the equation is:\[ x + 12y = 1.09 \]

Step 2: Solve the System of Equations

Subtract the second equation from the first to eliminate \( x \):\[ (x + 35y) - (x + 12y) = 2.93 - 1.09 \]This simplifies to:\[ 23y = 1.84 \]Solving for \( y \):\[ y = \frac{1.84}{23} = 0.08 \]Substitute \( y = 0.08 \) back into the second equation:\[ x + 12(0.08) = 1.09 \]\[ x + 0.96 = 1.09 \]\[ x = 0.13 \]

Step 3: Calculate Total Cost with Taxes

The problem involves tax rates: federal tax \(3.2\%\) and state tax \(7.2\%\), totaling \(10.4\%\). First, determine the pre-tax cost that would result in a final cost of \(\$5.00\):\[ C(1 + 0.104) = 5.00 \]Solving for \(C\) (the pre-tax cost):\[ C = \frac{5.00}{1.104} \approx 4.53 \]

Step 4: Determine Maximum Duration of Call

The cost equation for \( m \) minutes of calling is:\[ x + (m-1) \cdot y = 4.53 \]Substitute \( x = 0.13 \) and \( y = 0.08 \):\[ 0.13 + (m-1) \cdot 0.08 = 4.53 \]\[ (m-1) \cdot 0.08 = 4.40 \]\[ m-1 = \frac{4.40}{0.08} = 55 \]Therefore, \( m = 56 \).

Key Concepts

System of Equations

In the given problem, we're dealing with a system of equations. This method is used to find values for more than one unknown in a set of mathematical relationships. Imagine trying to solve a puzzle with two pieces: figuring out the cost of the first minute and each additional minute of a phone call.

• We use two different scenarios: a 36-minute call and a 13-minute call.
• We represent the unknowns as variables: let's call the cost of the first minute \( x \) and the cost for each additional minute \( y \).
• The equations formed from the two call scenarios are: \( x + 35y = 2.93 \) for 36 minutes and \( x + 12y = 1.09 \) for 13 minutes.

The purpose here is to solve these equations simultaneously, which means finding values for \( x \) and \( y \) that satisfy both equations at the same time.

Cost Analysis

Cost analysis involves breaking down the costs involved in specific situations. It's like understanding how much your phone bill can be before taxes! For this exercise, we're analyzing call costs by diving into the details of how much it costs per minute.

• First, find the cost of the initial minute and then each subsequent minute. This is important because each minute outside the first adds up differently.
• By solving the equations determined previously, we identified: the first minute costs 13 cents (\( x = 0.13 \)) and each additional minute costs 8 cents (\( y = 0.08 \)).

Understanding how these costs are calculated helps us determine the total cost for calls of any duration by combining these costs effectively.

Tax Calculation

Taxes often apply to many real-world situations, and here, they affect the total cost of a phone call. Understanding tax calculation is essential, as it reveals how the final cost is computed beyond just the base call prices.

• Two types of taxes are considered in this problem: federal tax at 3.2% and state tax at 7.2%.
• Combined, these taxes add up to a 10.4% extra charge on the base cost of the call.
• To find the pre-tax cost from a total of \$5.00, divide by 1.104 (which is 100% cost + 10.4% tax). This gives \( C = 4.53 \) for the maximum pre-tax cost.

This section shows how to convert between pre-tax and total costs, ensuring we stay within budget constraints while considering taxes.

Call Duration Calculation

Finally, determining the maximum call duration is like finding out how long you can talk before hitting a set limit, which in the problem was Established as \\(5.00 with taxes included. To calculate this duration while staying under the limit, we need to utilize what we've learned about costs.

• Using our costs for the first and additional minutes (\( x \) and \( y \)), and the pre-tax limit of \( 4.53 \), we derive the call duration equation: \( 0.13 + (m-1) \cdot 0.08 = 4.53 \).
• Simplifying gives us \( m-1 = \frac{4.40}{0.08} = 55 \).
• Finally, adding 1 back to \( m \) for the initial minute, we find \( m = 56 \). This means a 56-minute call would cost right up to \\)5.00 including taxes.

This highlights the importance of understanding the relationship between each additional component, allowing for precise budget management while making calls.