Problem 40 - Precalculus: Functions and Graphs

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 city a 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

The first minute costs \$0.13, additional minutes \$0.08. Max 56 min call under \$5.00.

Step 1: Formulate the equations

Let's call \( x \) the cost of the first minute and \( y \) the cost of each additional minute. We can set up two equations based on the information given in the problem for the 36-minute call and the 13-minute call.\\( x + 35y = 2.93 \) (Equation 1 for the 36-minute call)\\( x + 12y = 1.09 \) (Equation 2 for the 13-minute call)

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: \( 23y = 1.84 \). To find \( y \), divide both sides by 23:\\[ y = \frac{1.84}{23} = 0.08 \].Use this value of \( y \) in Equation 2 to find \( x \):\\[ x + 12(0.08) = 1.09 \] \[ x + 0.96 = 1.09 \] Thus, \( x = 0.13 \). Therefore, the cost of the first minute is \( \\(0.13 \) and each additional minute costs \( \\)0.08 \).

Step 3: Calculate the total tax rate

First, find the total tax rate by adding the federal and state tax rates: \( 3.2\% + 7.2\% = 10.4\% \) or \( 0.104 \) in decimal.

Step 4: Determine the maximum pre-tax cost of the call

Since the total (after tax) cost must not exceed \\(5.00, we need the pre-tax cost of the call: Let \( C \) be the cost before tax. Then, \[ C + 0.104C \leq 5.00 \] \[ 1.104C \leq 5.00 \] To find \( C \), divide both sides by 1.104: \[ C \leq \frac{5.00}{1.104} \approx 4.53 \] Therefore, the cost of the call before tax must not exceed \\)4.53.

Step 5: Calculate the longest call time allowable

Now, use the cost values \( x = 0.13 \) and \( y = 0.08 \) to find the maximum number of minutes.\( x + (m-1)y \leq 4.53 \), where \( m \) is the total minutes. \[ 0.13 + (m-1)(0.08) \leq 4.53 \] Solving for \( m \):\[ 0.08(m-1) \leq 4.40 \] \[ m-1 \leq \frac{4.40}{0.08} \] \[ m-1 \leq 55 \] \[ m \leq 56 \].Thus, the longest call that can be made without exceeding \$5.00 is 56 minutes.

Key Concepts

Linear Equations

A linear equation is an equation that makes a straight line when graphed. It is generally expressed in the form of \( ax + b = c \) where \( x \) represents the variable, and \( a \), \( b \), and \( c \) are constants. In this exercise, we are given two linear equations that represent the total cost of calls. Each equation is formed by setting the cost of the first minute \( x \) and the cost of each additional minute \( y \) to the total cost for each call.
These equations are:

• For a 36-minute call: \( x + 35y = 2.93 \)
• For a 13-minute call: \( x + 12y = 1.09 \)

Their role is to help us determine the unknown costs \( x \) and \( y \). By using methods like substitution or elimination, you can solve these equations to find values for \( x \) and \( y \), allowing you to calculate particular values based on given information. Linear equations play a crucial role in expressing relationships and solving for unknowns in many real-world applications.

Call Cost Calculation

Calculating the cost of a call involves considering both the initial and per-minute costs of the call and then applying these to the duration of the call. In this scenario, the call cost calculation begins with identifying two important values: the cost for the first minute \( x \) and the cost for each additional minute \( y \).
Here's how the calculation might work: consider a call's total cost as the sum of the charge for the first minute and the charges for the additional minutes. For example, in the 36-minute call, the cost formula looks like:

• Cost of Call = \( x + 35y \)

In the context of the problem, we found that for the 36-minute call, the call cost is \( 2.93 \) dollars. This is set equal to \( x + 35y \). Similarly, the cost calculation is done for the 13-minute call. By solving the system of equations derived from these calculations, we can find \( x = 0.13 \) and \( y = 0.08 \), where \( x \) represents the initial cost and \( y \) is the per minute cost thereafter.

Tax Rate Calculation

Calculating the tax rate involves summing the individual tax rates applicable to a cost or an item. In our scenario, we have a long-distance call that is subjected to both federal and state tax rates. It's crucial to determine the overall tax rate that will be applied to a call to find out the total cost including taxes.
Here, we have:

• Federal Tax Rate: \( 3.2\% \)
• State Tax Rate: \( 7.2\% \)

Thus, the total tax rate is the sum: \( 3.2\% + 7.2\% = 10.4\% \). This can be expressed as a decimal \( 0.104 \) in calculations. Understanding the total tax rate is important as it affects the final cost of the call, influencing budget decisions and cost estimations.

Pre-Tax Cost Estimation

Pre-tax cost estimation is about determining the cost of an item or service before any taxes are applied. It involves deciding what the maximum allowable amount is before taxes, given a certain budget limit after taxes are included.
In this problem, the customer does not want the final cost of the call to exceed \( \\(5.00 \). To find the maximum pre-tax cost, we solve:\[ C + 0.104C \leq 5.00 \]Simplifying gives:\[ 1.104C \leq 5.00 \]Thus, dividing both sides by \( 1.104 \), we find:\[ C \leq \frac{5.00}{1.104} \approx 4.53 \]This outcome tells us that before applying taxes, the cost of a call should not be more than \( \\)4.53 \).This approach ensures that taxes won’t push the cost over the set budget, allowing better financial planning.