A phone company has a monthly cellular data plan where a...

A phone company has a monthly cellular data plan where a customer pays a flat monthly fee and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses $20 \mathrm{MB},$ the monthly cost will be $\$ 11.20 .$ If the customer uses $130 \mathrm{MB}$, the monthly cost will be $\$ 17.80$. a. Find a linear equation for the monthly cost of the data plan as a function of $x$, the number of MB used. b. Interpret the slope and vertical intercept of the equation. c. Use your equation to find the total monthly cost if $250 \mathrm{MB}$ are used.

A phone company has a monthly cellular data plan where a customer pays a flat monthly fee and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses $20 \mathrm{MB},$ the monthly cost will be $\$ 11.20 .$ If the customer uses $130 \mathrm{MB}$, the monthly cost will be $\$ 17.80$.
a. Find a linear equation for the monthly cost of the data plan as a function of $x$, the number of MB used.
b. Interpret the slope and vertical intercept of the equation.
c. Use your equation to find the total monthly cost if $250 \mathrm{MB}$ are used.

Key Concepts

Linear Equation Modeling

Linear equation modeling is a technique used to represent a real-world relationship with a simple linear equation. This process involves determining both the fixed and variable components from given data points to build a model that can predict outcomes, such as total monthly costs based on data usage.

Variable Cost (Slope)

The slope in a cost function indicates the additional cost incurred for each extra unit of consumption. It provides the rate of change of the total cost with respect to usage, such as the charge per megabyte in a data plan, thereby illustrating how costs escalate with increased usage.

Linear Functions

A linear function is an equation that forms a straight line when graphed and is typically written in the form y = mx + b. It represents a constant rate of change between the dependent variable and the independent variable, allowing for straightforward predictions and interpretations in various real-world scenarios.

Cost Functions

Cost functions are specialized linear functions used to model pricing structures. They combine a fixed cost (a constant fee regardless of usage) and a variable cost (which increases in relation to usage), making them ideal for situations such as cellular data plans.

Fixed Cost (Vertical Intercept)

The fixed cost, represented by the vertical intercept in a linear equation, is the cost that does not change with the level of consumption. In many service plans, this is the base fee that a customer must pay even if no additional usage occurs.

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