Questions: The table below shows the cost, C, of selling x cups of coffee per day from a cart. Determine if the relationship is linear, and find the slope of the line. x 0 5 10 50 100 200 C 70.00 72.00 74.00 90.00 110.00 150.00 The relationship is , the slope of the line is (Enter your answer as a decimal).

Solution Steps

To determine if the relationship is linear, we need to check if the rate of change (slope) between each pair of points is constant. If the slope is constant, the relationship is linear. The slope between two points \((x_1, C_1)\) and \((x_2, C_2)\) can be calculated using the formula:

\[ \text{slope} = \frac{C_2 - C_1}{x_2 - x_1} \]

We will calculate the slope between each consecutive pair of points and check if they are all the same.

1. Extract the values of \(x\) and \(C\) from the table.
2. Calculate the slope between each consecutive pair of points.
3. Check if all calculated slopes are equal.
4. If they are equal, the relationship is linear and the slope is the common value. Otherwise, the relationship is not linear.

We are given the values of \(x\) and \(C\) from the table: \[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 5 & 10 & 50 & 100 & 200 \\ \hline C & 70.00 & 72.00 & 74.00 & 90.00 & 110.00 & 150.00 \\ \hline \end{array} \]

We calculate the slope between each consecutive pair of points using the formula: \[ \text{slope} = \frac{C_2 - C_1}{x_2 - x_1} \] The calculated slopes are: \[ \begin{align_} \text{slope}_{1} &= \frac{72.00 - 70.00}{5 - 0} = 0.4 \\ \text{slope}_{2} &= \frac{74.00 - 72.00}{10 - 5} = 0.4 \\ \text{slope}_{3} &= \frac{90.00 - 74.00}{50 - 10} = 0.4 \\ \text{slope}_{4} &= \frac{110.00 - 90.00}{100 - 50} = 0.4 \\ \text{slope}_{5} &= \frac{150.00 - 110.00}{200 - 100} = 0.4 \\ \end{align_} \]

Since all calculated slopes are equal (\(0.4\)), the relationship is linear.

Final Answer