He purchases a total of 150 containers of popcorn to sell. After he sells 15 containers, he will make a total profit, after his upfront costs, of \( \$ 5.50 \). Cassidy's profit, after upfront costs, can be modeled by a linear function in the form \( y=m x+b \).

To find Cassidy's profit after upfront costs, we can use the given information to determine the slope and y-intercept of the linear function. Given: - Total containers of popcorn purchased = 150 - Containers sold = 15 - Profit after selling 15 containers = $5.50 Let's denote the profit after upfront costs as \( y \) and the number of containers sold as \( x \). The profit after selling 15 containers can be represented as: \[ y = 5.50 \] We can use the point-slope form of a linear equation to find the slope (\( m \)) and y-intercept (\( b \)) of the linear function. The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] Substitute the given values into the point-slope form: \[ y - 5.50 = m(x - 15) \] Now, we need to find the slope (\( m \)) and y-intercept (\( b \)) of the linear function. Let's calculate the slope (\( m \)) first. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(5.5-0\right)}{\left(15-0\right)}\) - step1: Remove the parentheses: \(\frac{5.5-0}{15-0}\) - step2: Remove 0: \(\frac{5.5}{15-0}\) - step3: Remove 0: \(\frac{5.5}{15}\) - step4: Convert the expressions: \(\frac{\frac{11}{2}}{15}\) - step5: Multiply by the reciprocal: \(\frac{11}{2}\times \frac{1}{15}\) - step6: Multiply the fractions: \(\frac{11}{2\times 15}\) - step7: Multiply: \(\frac{11}{30}\) The slope (\( m \)) of the linear function is \( \frac{11}{30} \) or approximately 0.3667. Now, let's find the y-intercept (\( b \)) of the linear function. Calculate the value by following steps: - step0: Calculate: \(5.5-\frac{11}{30}\times 15\) - step1: Multiply the numbers: \(5.5-\frac{11}{2}\) - step2: Convert the expressions: \(\frac{11}{2}-\frac{11}{2}\) - step3: Transform the expression: \(\frac{11-11}{2}\) - step4: Subtract the numbers: \(\frac{0}{2}\) - step5: Calculate: \(0\) The y-intercept (\( b \)) of the linear function is 0. Therefore, the linear function that models Cassidy's profit after upfront costs is: \[ y = \frac{11}{30}x \] This linear function represents the profit Cassidy will make after selling a certain number of containers of popcorn, taking into account the upfront costs.