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 5th commercial. Convert your answer to dollars. The cost to air the 5 th commercial is approximately (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 solve the problem, we will go through each part step by step. ### (a) Find the marginal cost function \( C^{\prime}(x) \). The cost function is given by: \[ C(x) = 150 + 2400x - 0.08x^2 \] To find the marginal cost function \( C^{\prime}(x) \), we need to take the derivative of \( C(x) \) with respect to \( x \): \[ C^{\prime}(x) = \frac{d}{dx}(150) + \frac{d}{dx}(2400x) + \frac{d}{dx}(-0.08x^2) \] Calculating each term: - The derivative of \( 150 \) is \( 0 \). - The derivative of \( 2400x \) is \( 2400 \). - The derivative of \( -0.08x^2 \) is \( -0.16x \). Putting it all together: \[ C^{\prime}(x) = 0 + 2400 - 0.16x = 2400 - 0.16x \] ### (b) Use the marginal cost to approximate the cost to air the 5th commercial. The marginal cost function \( C^{\prime}(x) \) gives us the approximate cost of airing the \( x \)-th commercial. To find the approximate cost to air the 5th commercial, we evaluate \( C^{\prime}(4) \): \[ C^{\prime}(4) = 2400 - 0.16(4) \] Calculating: \[ C^{\prime}(4) = 2400 - 0.64 = 2399.36 \] Since the cost is given in thousands of dollars, we convert this to dollars: \[ \text{Cost to air the 5th commercial} \approx 2399.36 \times 1000 = 2399360 \text{ dollars} \] ### (c) What is the exact cost to air the 5th commercial? To find the exact cost to air the 5th commercial, we need to calculate the cost of airing 5 commercials and subtract the cost of airing 4 commercials: \[ \text{Cost to air 5 commercials} = C(5) \] \[ \text{Cost to air 4 commercials} = C(4) \] Calculating \( C(5) \): \[ C(5) = 150 + 2400(5) - 0.08(5^2) \] \[ C(5) = 150 + 12000 - 0.08(25) \] \[ C(5) = 150 + 12000 - 2 = 11848 \text{ (in thousands of dollars)} \] Calculating \( C(4) \): \[ C(4) = 150 + 2400(4) - 0.08(4^2) \] \[ C(4) = 150 + 9600 - 0.08(16) \] \[ C(4) = 150 + 9600 - 1.28 = 10148 \text{ (in thousands of dollars)} \] Now, the exact cost to air the 5th commercial is: \[ C(5) - C(4) = 11848 - 10148 = 1700 \text{ (in thousands of dollars)} \] Converting this to dollars: \[ \text{Exact cost to air the 5th commercial} = 1700 \times 1000 = 1700000 \text{ dollars} \] ### Summary of Answers: (b) The cost to air the 5th commercial is approximately **$2,399,360**. (c) The exact cost to air the 5th commercial is **$1,700,000**.