Worldwide annual sales of all cellphones were approximately
−10p + 1,600 million phones
when the wholesale price was $p. At what wholesale price should cellphones have been sold to maximize annual revenue?
To find the wholesale price at which cellphones should have been sold to maximize annual revenue, we need to determine the revenue function and then find its maximum value.
The revenue is calculated by multiplying the unit price by the number of cellphones sold. Let's denote the unit price as "p" (wholesale price), and the number of cellphones sold as "x".
The revenue function can be expressed as follows:
Revenue = Unit Price * Number of Cellphones Sold
Revenue = p * x
Given that the number of cellphones sold is approximately -10p + 1,600 million phones, we can substitute this expression for "x" in the revenue function:
Revenue = p * (-10p + 1,600 million)
To find the maximum revenue, we can take the derivative of the revenue function with respect to the wholesale price "p" and set it equal to zero.
dRevenue/dp = -10p + 1,600 million = 0
Solving this equation will give us the value of "p" at which the revenue is maximized.
-10p + 1,600 million = 0
-10p = -1,600 million
p = (-1,600 million) / (-10)
p = 160 million
Therefore, cellphones should have been sold at a wholesale price of $160 million to maximize annual revenue.
To find the wholesale price at which cellphones should have been sold to maximize annual revenue, we need to determine the revenue function. Revenue is calculated by multiplying the quantity of phones sold by the price at which they are sold.
Let's start by representing the quantity of phones sold as a function of the wholesale price using the given equation:
Quantity (Q) = -10p + 1,600 million phones
Next, we need to express the revenue function in terms of p. We can do this by multiplying the quantity (Q) by the wholesale price (p):
Revenue (R) = p * Q
Revenue (R) = p * (-10p + 1,600 million)
To maximize the annual revenue, we need to find the wholesale price (p) that maximizes the revenue function (R). One way to do this is by finding the vertex of the revenue function, which corresponds to the maximum point.
The revenue function is a quadratic equation, in the form of:
R = -10p^2 + 1,600 million p
To find the vertex, we can use the formula:
p = -b / (2a)
In this case, a = -10 and b = 1,600 million. Substituting these values into the formula, we can find p:
p = -(1,600 million) / (2 * -10)
p = -(1,600 million) / (-20)
p = 80 million
Therefore, cellphones should have been sold at a wholesale price of $80 million to maximize annual revenue.