Final answer:
The two positive numbers that maximize the product while their sum and twice the second is 280 are 140 and 70.
Step-by-step explanation:
To find two positive numbers that satisfy the requirements, let's call the first number x and the second number y. The problem states that the sum of the first and twice the second is 280, which gives us the equation x + 2y = 280.
To maximize the product of the two numbers, we can use the fact that for a fixed sum, the product of two numbers is maximized when the numbers are equal, or as close to equal as possible due to the arithmetic mean-geometric mean inequality.
Therefore, we first express y in terms of x from the first equation: y = (280 - x)/2. Then we write the product P = x × y and substitute y with the expression in terms of x to get P = x(280 - x)/2. This function can be maximized by finding its vertex, since it represents a parabola that opens downwards. Differentiating with respect to x and setting the derivative equal to 0, we find x = 140, and consequently, y = 70.
Hence the two numbers are 140 and 70, which indeed give the maximum product under the given constraint.