1ppose a calculator manufacturer has the total cost function \( C(x)=38 x+6400 \) and the total revenue function \( R(x)=70 x \). (a) What is the equation of the profit function \( P(x) \) for the calculator? \( P(x)= \) (b) What is the profit on 2400 units? \( P(2400)=\$ \)

To find the profit function \( P(x) \), we start with the definitions of total cost \( C(x) \) and total revenue \( R(x) \). The profit function is given by the formula: \[ P(x) = R(x) - C(x) \] Given: - Total cost function: \( C(x) = 38x + 6400 \) - Total revenue function: \( R(x) = 70x \) Now, we can substitute these functions into the profit function equation: \[ P(x) = R(x) - C(x) = 70x - (38x + 6400) \] Simplifying this, we get: \[ P(x) = 70x - 38x - 6400 \] \[ P(x) = 32x - 6400 \] So, the equation of the profit function is: \[ \boxed{P(x) = 32x - 6400} \] Next, we need to calculate the profit when \( x = 2400 \): \[ P(2400) = 32(2400) - 6400 \] Calculating \( 32(2400) \): \[ 32 \times 2400 = 76800 \] Now substituting this back into the profit function: \[ P(2400) = 76800 - 6400 \] \[ P(2400) = 70400 \] Thus, the profit on 2400 units is: \[ \boxed{70400} \]