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 of 10 dollar 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 dollar. If the customer uses $130 \mathrm{MB},$ the monthly cost will be 17.80 dollar. a. Find a linear equation for the monthly cost of the data plan as a function of $x,$ the number of $\mathrm{MB}$ used. b. Interpret the slope and $y$ -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 of 10 dollar 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 dollar. If the customer uses
$130 \mathrm{MB},$ the monthly cost will be 17.80 dollar.
a. Find a linear equation for the monthly cost of the data plan as a function of $x,$ the number of $\mathrm{MB}$ used.
b. Interpret the slope and $y$ -intercept of the equation.
c. Use your equation to find the total monthly cost if
$250 \mathrm{MB}$ are used.

Key Concepts

Slope-Intercept Form

The slope-intercept form, y = mx + b, is a commonly used representation of a linear function. In this format, 'm' denotes the rate at which the dependent variable changes with respect to the independent variable, while 'b' indicates the value of the dependent variable when the independent variable is zero.

Interpreting the Slope

The slope in a linear equation quantifies the change in the dependent variable for each unit increase in the independent variable. It represents the constant rate at which one variable responds to changes in the other, making it key for understanding the relationship between the two quantities.

Linear Equations

Linear equations express a constant rate of change between two quantities. They form a straight-line graph when plotted on a coordinate plane and are typically written in the form y = mx + b, where m represents the slope and b the y-intercept.

Constructing a Linear Equation from Two Data Points

Given two pairs of corresponding values, one can determine the linear relationship by calculating the slope (rate of change) and then using one of the points to solve for the y-intercept. This method is essential in creating a model that accurately represents the relationship between the variables.

Interpreting the Y-Intercept

The y-intercept is the point where the linear graph crosses the y-axis, meaning it shows the initial value of the dependent variable when the independent variable is zero. This value often represents a fixed or starting cost before any variable costs are added.

Using Linear Models for Predictions

Once a linear equation is formulated, it can be used to predict outcomes for any given value of the independent variable within the model's range. This predictive capability is crucial for estimating results and making informed decisions based on the established relationship.

Transcript

00:01 In this problem, we have a cell phone company that's charging a flat rate plus a charge per megabyte of data.

00:08 So we know that at 20 megabytes of data, the cost is 1120, and at 13 megabytes, the cost is 1780.

00:17 So let's define x as my cost per megabyte.

00:22 Okay.

00:24 And we're going to solve for that.

00:26 So in the first situation, i know that if i have, um, 20 megabytes times some cost plus $10 is going to equal to 1120.

00:42 And i need to solve for x.

00:44 So to do that, i would subtract 10 from both sides.

00:47 Remember, you want your variables on one side, your numbers on the other.

00:52 So when i do that, i'm left with 20x is equal to 120.

01:02 Now i'm going to divide both sides by 20.

01:04 And then 1 .2 divided by 20 gives me x equals 0 .06.

01:16 Okay.

01:17 So that tells me that they're charging $0 .6 per megabyte.

01:22 Now, let's just double check here.

01:23 Let's go ahead and check the other scenario just to make sure it's the same amount.

01:29 So at 130 megabytes times some rate plus 10, that was 1780.

01:36 So we would do the same thing.

01:38 We would subtract 10, subtract 10.

01:42 So that would give me 130 x is equal to 1780 minus 10 is 780.

01:52 And then we come along and divide both sides by 130, divide by 130.

01:58 And i get that x is equal to 0 .06.

02:03 Okay, so now the first step in this problem is they wanted us to write be linear equation.

02:10 So linear equations are in slope intercept form.

02:15 Okay, so in this case, y equals mx plus b, where m is your slope or your rate of change, and then b is your initial cost or your starting point...

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