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A concert promoter sells tickets and has a marginal-profit function given below, where P(x) is in dollars per ticket. This means that the rate of change of total profit with respect to the number of tickets sold, x, is P'(X). Find the total profit from the sale of the first 70 tickets, disregarding any foxed costs. P' (*)=7x-1101 The total profit is $ (Round to the nearest cent as needed.) 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 60 lb of coffee, disregarding any fixed costs. C' O)=-0.021x* 3.25, for x5 100 The total cost is $ (Round to the nearest cent as needed)

Answer :

The total profit from selling the first 70 concert tickets, based on the given marginal-profit function, is $18,049.20. The total cost of roasting 60 pounds of coffee, based on the provided marginal cost function, is $3,511.92.

To find the total profit from the sale of the first 70 concert tickets, we need to integrate the marginal-profit function P'(x) = 7x - 1101 with respect to x from 0 to 70. Integrating the function gives us the total profit function P(x) = (7/2)x^2 - 1101x + C, where C is the constant of integration. To find C, we can substitute the given information that P(0) = 0 (no profit when no tickets are sold). By solving this equation, we find C = 0. Thus, the total profit function becomes P(x) = (7/2)x^2 - 1101x.

To find the total profit from the sale of the first 70 tickets, we evaluate P(70). Plugging in x = 70 into the total profit function gives us P(70) = (7/2)(70)^2 - 1101(70) = $18,049.20. Therefore, the total profit from the sale of the first 70 tickets is $18,049.20.

Moving on to the coffee company's total cost of roasting 60 pounds of coffee, we are given the marginal cost function C'(x) = -0.021x * 3.25 for x ≥ 100. This function represents the cost per pound of coffee roasted. To find the total cost, we need to integrate the marginal cost function with respect to x from 100 to 60 (since x represents the number of pounds of coffee roasted). Integrating the function gives us the total cost function C(x) = -0.021x^2 * 3.25/2 + C, where C is the constant of integration. To find C, we can use the given information that C(100) = 0 (no cost when 100 pounds are roasted). By solving this equation, we find C = 0. Thus, the total cost function becomes C(x) = -0.021x^2 * 3.25/2.

To find the total cost of roasting 60 pounds of coffee, we evaluate C(60). Plugging in x = 60 into the total cost function gives us C(60) = -0.021(60)^2 * 3.25/2 = $3,511.92. Therefore, the total cost of roasting 60 pounds of coffee, disregarding any fixed costs, is $3,511.92.

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