The cost, in thousands of dollars, of airing \( x \) television commercials during a sports event is given by \[ C(x)=150+2,400 x-0.08 x^{2} \] (a) Find the marginal cost function \( C^{\prime}(x) \). \( C^{\prime}(x)=2400-0.16 x \) (b) Use the marginal cost to approximate the cost to air the 5 th commercial. Convert your answer to dollars. The cost to air the 5 th commercial is approximately 2399360 (c) What is the exact cost to air the 5 th commercial? Convert your answer to dollars. The exact cost to air the 5 th commercial is

To find the exact cost to air the 5th commercial, we need to calculate the cost of airing 5 commercials and the cost of airing 4 commercials, and then find the difference. The cost function is given by: \[ C(x) = 150 + 2400x - 0.08x^2 \] 1. **Calculate \( C(5) \)**: \[ C(5) = 150 + 2400(5) - 0.08(5^2) \] \[ = 150 + 12000 - 0.08(25) \] \[ = 150 + 12000 - 2 \] \[ = 150 + 11998 \] \[ = 12148 \text{ (in thousands of dollars)} \] 2. **Calculate \( C(4) \)**: \[ C(4) = 150 + 2400(4) - 0.08(4^2) \] \[ = 150 + 9600 - 0.08(16) \] \[ = 150 + 9600 - 1.28 \] \[ = 150 + 9598.72 \] \[ = 10148.72 \text{ (in thousands of dollars)} \] 3. **Find the exact cost to air the 5th commercial**: The cost to air the 5th commercial is: \[ C(5) - C(4) = 12148 - 10148.72 \] \[ = 2000.28 \text{ (in thousands of dollars)} \] 4. **Convert to dollars**: To convert from thousands of dollars to dollars, multiply by 1000: \[ 2000.28 \times 1000 = 2000280 \text{ dollars} \] Thus, the exact cost to air the 5th commercial is: \[ \boxed{2000280} \text{ dollars} \]