The cost function for a product is \( C(x)=0.6 x^{2}+110 x+100 \). Find average cost over \( [0,500] \). Answer:

To find the average cost over the interval \([0, 500]\), we first need to calculate the total cost over that interval and then divide by the length of the interval. The average cost \( A \) over the interval \([a, b]\) is given by the formula: \[ A = \frac{1}{b - a} \int_a^b C(x) \, dx \] In this case, \( a = 0 \) and \( b = 500 \). Thus, we need to compute: \[ A = \frac{1}{500 - 0} \int_0^{500} C(x) \, dx = \frac{1}{500} \int_0^{500} (0.6 x^2 + 110 x + 100) \, dx \] Now, we will calculate the integral: \[ \int (0.6 x^2 + 110 x + 100) \, dx = \frac{0.6}{3} x^3 + \frac{110}{2} x^2 + 100 x + C = 0.2 x^3 + 55 x^2 + 100 x + C \] Next, we evaluate this integral from \(0\) to \(500\): \[ \int_0^{500} (0.6 x^2 + 110 x + 100) \, dx = \left[ 0.2 x^3 + 55 x^2 + 100 x \right]_0^{500} \] Calculating at \(x = 500\): \[ 0.2 (500)^3 + 55 (500)^2 + 100 (500) \] Calculating each term: 1. \(0.2 (500)^3 = 0.2 \times 125000000 = 25000000\) 2. \(55 (500)^2 = 55 \times 250000 = 13750000\) 3. \(100 (500) = 50000\) Now, summing these values: \[ 25000000 + 13750000 + 50000 = 38700000 \] Now, we evaluate at \(x = 0\): \[ 0.2 (0)^3 + 55 (0)^2 + 100 (0) = 0 \] Thus, the definite integral from \(0\) to \(500\) is: \[ \int_0^{500} (0.6 x^2 + 110 x + 100) \, dx = 38700000 - 0 = 38700000 \] Now, we can find the average cost: \[ A = \frac{1}{500} \times 38700000 = 77400 \] Therefore, the average cost over the interval \([0, 500]\) is: \[ \boxed{77400} \]