Questions: 3. A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of 10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be 11.20. If the customer uses 130 MB, the monthly cost will be 17.80. (10 points) a) Find a linear equation for the monthly cost of the data plan as a function of x, the number of MB used. b) Use your equation to find the total monthly cost if 250 MB are used.

Solution Steps

To solve this problem, we need to determine the linear equation that represents the monthly cost as a function of the number of megabytes (MB) used. We can use the given data points to find the slope and y-intercept of the linear equation.

1. Identify the two data points: (20, 11.20) and (130, 17.80).
2. Calculate the slope (m) using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
3. Use one of the points and the slope to find the y-intercept (b) using the formula: \( y = mx + b \).
4. Formulate the linear equation: \( y = mx + b \).
5. Use the linear equation to calculate the total monthly cost for 250 MB.

Using the two data points \((20, 11.20)\) and \((130, 17.80)\), we calculate the slope \(m\) as follows:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{17.80 - 11.20}{130 - 20} = \frac{6.60}{110} = 0.0600 \]

Next, we find the y-intercept \(b\) using the point \((20, 11.20)\):

\[ b = y_1 - m \cdot x_1 = 11.20 - 0.0600 \cdot 20 = 11.20 - 1.20 = 10.0 \]

The linear equation for the monthly cost \(C\) as a function of the number of megabytes used \(x\) is:

\[ C(x) = 0.0600x + 10.0 \]

To find the total monthly cost when \(x = 250\) MB, we substitute \(250\) into the linear equation:

\[ C(250) = 0.0600 \cdot 250 + 10.0 = 15.00 + 10.0 = 25.00 \]

Final Answer