Final answer:
The total cost of making 20 toys is 45. The total revenue of selling 10 toys is 8970. The total profit function is P(x) = x(947-5x) - (2x+5). The total profit of making and selling 11 toys is 9785. To break even, 189 toys need to be made.
Step-by-step explanation:
To find the total cost of making 20 toys, we can substitute x=20 into the cost function C(x)=2x+5. Therefore, the total cost is C(20) = 2(20) + 5 = 40 + 5 = 45.
To find the total revenue of selling 10 toys, we can substitute x=10 into the revenue function R(x)=x(947-5x). Therefore, the total revenue is R(10) = 10(947-5(10)) = 10(947-50) = 10(897) = 8970.
The total profit function P(x) is obtained by subtracting the total cost function from the total revenue function. Therefore, P(x) = R(x) - C(x) = x(947-5x) - (2x+5).
To find the total profit of making and selling 11 toys, we can substitute x=11 into the profit function P(x). Therefore, the total profit is P(11) = 11(947-5(11)) - (2(11)+5) = 11(947-55) - (22+5) = 11(892) - 27 = 9812 - 27 = 9785.
To find the break-even point, we need to find the quantity of toys where the total revenue is equal to the total cost. Set R(x) = C(x) and solve for x. Since R(x) = x(947-5x) and C(x) = 2x+5, we have x(947-5x) = 2x+5. Rearranging the equation gives 947x - 5x^2 = 2x+5. Simplifying further results in 5x^2 - 945x + 5 = 0. Using the quadratic formula, we can solve for x. Plugging in the values into the quadratic formula gives x = 189 or x = 0. Therefore, we need to make 189 toys in order to break even.