Final answer:
The maximum revenue for the company is achieved by setting the price of the phone at $204. This is determined by finding the vertex of the revenue function, which is a quadratic equation that opens downwards.
Step-by-step explanation:
The student is asking about maximizing a company's revenue based on the price of a product, specifically phones in this case. The given function for the number of phones sold is f(x) = 4080 - 10x, where x is the price set for the phones. To maximize the company's revenue, which is x × (4080 - 10x), we need to find the value of x that will give the highest revenue.
To find the price x that maximizes revenue, we can use calculus by taking the derivative of the revenue function and setting it equal to zero to find critical points. Alternatively, we can recognize that this is a quadratic function that opens downwards (since the x^2 term will have a negative coefficient), which means that the vertex of the parabola will represent the maximum point. The formula for finding the x-value of the vertex of a parabola given by ax^2 + bx + c is -b/(2a). For the revenue function x × (4080 - 10x) or -10x^2 + 4080x, a is -10 and b is 4080. Using the vertex formula, we find the price x to be -4080/(2×(-10)) = 204.