A company charting its profits notices that the relationsthip between the mumber of units sold, s, and the profit, P, is linear. If 190 units sold results in \( \$ 2380 \) profit and 240 units sold results in \( \$ 4980 \) profit, write the profit function for this company. \( P=\square \) Find the marginal profit. \( \$ \square \)

To find the profit function \( P(s) \) that relates the number of units sold \( s \) to the profit \( P \), we can use the information given about the two points: 1. When \( s = 190 \), \( P = 2380 \) 2. When \( s = 240 \), \( P = 4980 \) Since the relationship is linear, we can express it in the form: \[ P(s) = ms + b \] where \( m \) is the slope and \( b \) is the y-intercept. ### Step 1: Calculate the slope \( m \) The slope \( m \) can be calculated using the formula: \[ m = \frac{P_2 - P_1}{s_2 - s_1} \] Substituting the values: \[ m = \frac{4980 - 2380}{240 - 190} = \frac{2600}{50} = 52 \] ### Step 2: Use one of the points to find \( b \) Now that we have the slope \( m = 52 \), we can use one of the points to find \( b \). Let's use the point \( (190, 2380) \): \[ P(190) = 52(190) + b \] \[ 2380 = 9880 + b \] \[ b = 2380 - 9880 = -7500 \] ### Step 3: Write the profit function Now we can write the profit function: \[ P(s) = 52s - 7500 \] ### Step 4: Find the marginal profit The marginal profit is the slope of the profit function, which we have already calculated as \( m = 52 \). Therefore, the marginal profit is: \[ \$ 52 \] ### Final Answers The profit function is: \[ P = 52s - 7500 \] The marginal profit is: \[ \$ 52 \]