Final answer:
To find the maximum profit for the new headphones, calculate the vertex of the profit function P(x), obtained by subtracting the cost function C(x) from the revenue function R(x). The profit max occurs at the x-value given by the formula -b/(2a), where b and a are coefficients from P(x).
Step-by-step explanation:
To determine which statements are true regarding the electronics store chain's new headphone model, we need to calculate the profit function, which is obtained by subtracting the total cost C(x) from the total revenue R(x). Thus, the profit function P(x) is:
P(x) = R(x) - C(x) = (-210x2 + 6,970x) - (-170x + 39,690)
Simplifying this equation gives us:
P(x) = -210x2 + 6,970x + 170x - 39,690
P(x) = -210x2 + 7,140x - 39,690
To find the maximum profit, we need to find the vertex of the parabola represented by the profit function. The x-coordinate of the vertex is given by -b/(2a), where a and b are the coefficients of x2 and x respectively. For the profit function P(x), a = -210 and b = 7,140, so:
x = -7140 / (2 * -210) = 7140 / 420 = 17
This means the maximum profit occurs when x, the number of units sold, is 17. Now we need to find the corresponding P(x) value to determine the maximum profit:
P(17) = -210(17)2 + 7,140(17) - 39,690
Calculating the value yields the maximum profit. However, we must also consider the options given for the selling price. The selling price per unit can be found by dividing the total revenue by the number of units sold. From the revenue function, R(x) = -210x2 + 6,970x, we can calculate the selling price that corresponds to the quantity of 17.
Once we determine the selling price that yields the maximum profit and the value of the maximum profit itself, we can identify which statements are true from the options provided.