Consider the marginal cost function \( \mathrm{C}^{\prime}(\mathrm{x})=200+12 \mathrm{x}-0.01 \mathrm{x}^{2} \) a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units. b. Find the additional cost incurred in dollars when production is increased from 400 units to 450 units.
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To solve the problems, we need to find the total cost function from the marginal cost function and then evaluate it over the given intervals. a. First, we need to integrate the marginal cost function \( C'(x) \): \[ C(x) = \int (200 + 12x - 0.01x^2) \, dx = 200x + 6x^2 - \frac{0.01}{3}x^3 + C_0 \] Now we will evaluate the total cost at \( x = 150 \) and \( x = 100 \): \[ C(150) = 200(150) + 6(150)^2 - \frac{0.01}{3}(150)^3 + C_0 \] Calculating \( C(150) \): \[ C(150) = 30000 + 13500 - \frac{0.01}{3}(3375000) + C_0 = 30000 + 13500 - 11250 + C_0 = 32250 + C_0 \] Now calculate \( C(100) \): \[ C(100) = 200(100) + 6(100)^2 - \frac{0.01}{3}(100)^3 + C_0 \] Calculating \( C(100) \): \[ C(100) = 20000 + 6000 - \frac{0.01}{3}(1000000) + C_0 = 20000 + 6000 - \frac{10000}{3} + C_0 \approx 26000 - 3333.33 + C_0 = 22666.67 + C_0 \] Now to find the additional cost incurred when producing from 100 to 150 units: \[ \text{Additional Cost} = C(150) - C(100) = (32250 + C_0) - (22666.67 + C_0) = 32250 - 22666.67 = 9583.33 \text{ dollars} \] b. Now, we will find the additional cost incurred when production is increased from 400 units to 450 units. We calculate \( C(450) \) and \( C(400) \): Calculating \( C(450) \): \[ C(450) = 200(450) + 6(450)^2 - \frac{0.01}{3}(450)^3 + C_0 \] Calculating \( C(450) \): \[ C(450) = 90000 + 12150 - \frac{0.01}{3}(91125000) + C_0 = 90000 + 12150 - 30375 + C_0 = 71775 + C_0 \] Now calculating \( C(400) \): \[ C(400) = 200(400) + 6(400)^2 - \frac{0.01}{3}(400)^3 + C_0 \] Calculating \( C(400) \): \[ C(400) = 80000 + 9600 - \frac{0.01}{3}(64000000) + C_0 = 80000 + 9600 - 21333.33 + C_0 = 68366.67 + C_0 \] Now to find the additional cost incurred when producing from 400 to 450 units: \[ \text{Additional Cost} = C(450) - C(400) = (71775 + C_0) - (68366.67 + C_0) = 71775 - 68366.67 = 3408.33 \text{ dollars} \] In summary, the additional cost incurred when increasing production from 100 to 150 units is approximately \( 9583.33 \) dollars, and when increasing from 400 to 450 units it is approximately \( 3408.33 \) dollars.
