a coffee company has found that the marginal cost, in dollars per pound, of the coffee it roasts is…
a coffee company has found that the marginal cost, in dollars per pound, of the coffee it roasts is represented by the function below, where x is the number of pounds of coffee roasted. find the total cost of roasting 250 lb of coffee, disregarding any fixed costs. c(x) = -0.016x + 8.50, for x ≤ 500
Answer
Explanation:
Step1: Recall the formula for total cost from marginal cost
The total cost (C(x)) (when fixed cost (C(0) = 0)) is given by (C(x)=\int_{0}^{x}C^{\prime}(t)dt). Here, (C^{\prime}(x)=0.016x + 8.50) and (x = 250). So, (C(250)=\int_{0}^{250}(0.016t+8.50)dt).
Step2: Integrate the marginal - cost function
Use the power rule (\int(ax + b)dx=\frac{ax^{2}}{2}+bx + C). (\int_{0}^{250}(0.016t + 8.50)dt=\left[0.016\times\frac{t^{2}}{2}+8.50t\right]_{0}^{250}). First, evaluate (0.016\times\frac{t^{2}}{2}+8.50t) at (t = 250): (0.016\times\frac{250^{2}}{2}+8.50\times250). (0.008\times62500+2125). (500 + 2125). Then, evaluate (0.016\times\frac{t^{2}}{2}+8.50t) at (t = 0): (0.016\times\frac{0^{2}}{2}+8.50\times0=0). Subtract the lower - limit value from the upper - limit value: ((500 + 2125)-0).
Answer:
The total cost of roasting (250) pounds of coffee is ($2625).