For an international telephone call, a telephone company charges \$0.43 for the first minute, \$0.32 for each additional minute, and a \$2.10 service charge. If the cost of a call is \$5.73, how long did the person talk?

Verified step by step guidance

1. Define the variable for the total time of the call in minutes. Let \(t\) represent the total number of minutes talked.
2. Write an expression for the total cost of the call. The cost consists of three parts: the first minute at \$0.43, the additional minutes at \$0.32 each, and a fixed service charge of \$2.10. So, the cost can be expressed as: \$0.43 + 0.32(t - 1) + 2.10$.
3. Set up an equation by equating the total cost expression to the given cost of \$5.73: \$0.43 + 0.32(t - 1) + 2.10 = 5.73$.
4. Simplify the equation by combining like terms and isolating the variable term: first combine \$0.43\( and \)2.10\(, then subtract this sum from both sides to isolate \)0.32(t - 1)$.
5. Solve for \(t\) by dividing both sides by \$0.32$ and then adding 1 to find the total number of minutes talked.

Key Concepts

Linear Equations

A linear equation represents a relationship between variables with a constant rate of change. In this problem, the total cost depends linearly on the number of minutes talked, allowing us to set up an equation to solve for the unknown time.

Piecewise Cost Structure

The cost is divided into different parts: a fixed service charge, a fixed rate for the first minute, and a different rate for additional minutes. Understanding how to separate and combine these parts is essential to model the total cost accurately.

Solving for an Unknown Variable

After forming the equation based on the cost structure, solving for the unknown variable (the total time talked) involves isolating the variable using algebraic operations. This step finds the exact duration of the call.