Jose a computer manufacturer has the total cost function \( C(x)=74 x+3600 \) (in dollars) and the total revenue function \( R(x)=374 x \) (in dollat (a) What is the equation of the profit function \( P(x) \) (in dollars) for this commodity? \( P(x)=\square \) (b) What is the profit (in dollars) on 360 items? \( P(360)=\$ \square \) (c) How many items must be sold to avoid losing money? \( x=\square \)

To solve the problem, we will first derive the profit function from the given cost and revenue functions. ### (a) Profit Function \( P(x) \) The profit function \( P(x) \) is defined as the total revenue \( R(x) \) minus the total cost \( C(x) \): \[ P(x) = R(x) - C(x) \] Given: - \( C(x) = 74x + 3600 \) - \( R(x) = 374x \) Substituting these into the profit function: \[ P(x) = 374x - (74x + 3600) \] Now, simplify the equation: \[ P(x) = 374x - 74x - 3600 \] \[ P(x) = 300x - 3600 \] So, the equation of the profit function is: \[ P(x) = 300x - 3600 \] ### (b) Profit on 360 items To find the profit when \( x = 360 \): \[ P(360) = 300(360) - 3600 \] Calculating \( 300 \times 360 \): \[ 300 \times 360 = 108000 \] Now, substituting back into the profit function: \[ P(360) = 108000 - 3600 \] \[ P(360) = 104400 \] Thus, the profit on 360 items is: \[ P(360) = \$104400 \] ### (c) Items to avoid losing money To avoid losing money, the profit must be greater than or equal to zero: \[ P(x) \geq 0 \] Setting the profit function to zero: \[ 300x - 3600 \geq 0 \] Solving for \( x \): \[ 300x \geq 3600 \] \[ x \geq \frac{3600}{300} \] \[ x \geq 12 \] Thus, the number of items that must be sold to avoid losing money is: \[ x = 12 \] ### Summary of Answers (a) \( P(x) = 300x - 3600 \) (b) \( P(360) = \$104400 \) (c) \( x = 12 \)